for any \(r\in (0,R]\)
\(\lim \limits _{j\rightarrow \infty }\mathcal {F}_h(A_{h_j},u_{h_j}; Q_r(x_0)) =\int _{ Q_r(x_0)} x)e(u):e(u)\,dx.\) 

Without loss of generality, we suppose \(R=1\)  and \(x_0=0.\) 
Let
\(c_{1,h}:=\inf \limits _{(x,\nu )\in Q_1\times \mathbb {S}^1} \varphi _h(x,\nu ),\qquad c_{2,h}:=\sup \limits _{(x,\nu )\in Q_1\times \mathbb {S}^1} \varphi _h(x,\nu );\) 
by (REF ) we have \(d_5c_{2,h}\le c_{1,h}.\) 
Since
\(\sup _h \mathcal {H}^1( Q_1\cap \partial A_h )< \infty ,\) 
by Lemma REF  for every \(h\ge 1\)  there exists \(\xi _h\in (0,1)^2\)  such that
\(\mathcal {H}^1(\lbrace y\in Q_1\cap \partial A_h:\,\, \text{${\rm tr}_{A_h}(u_h) $ exists andequals to $\xi _h$ at $y$}\rbrace )=0.\) 
Therefore
\(\widetilde{u}_h: ={\left\lbrace \begin{array}{ll}u_h & \text{in $ Q_1\cap A_h,$}\\\xi _h & \text{in $ Q_1\setminus A_h$}\end{array}\right.}\) 
belongs to \(GSBD^2( Q_1;\mathbb {R}^2)\)  with
\(J_{\widetilde{u}_h}\subset Q_1\cap \partial A_h\) 
and
\(\lim \limits _{h\rightarrow \infty } \mathcal {H}^1(J_{\widetilde{u}_h}) = 0\) 
in view of (). Further we suppose \(\mathcal {H}^1(J_{\widetilde{u}_h})<1/4\)  for any \(h\ge 1.\)
By [1]} and (REF ), there exist a constant \(c\)  (depending only on \(d_3\) ) and sequences \(\lbrace \widetilde{\omega }_h\rbrace \)  of a Lebesgue measurable subsets \( Q_1\)  with \(|\widetilde{\omega }_h|\le c\mathcal {H}^1( Q_1\cap \partial A_h )\)  and \(\lbrace a_h\rbrace \)  of rigid motions such that
\(\int _{ Q_1\setminus \widetilde{\omega }_h} |\widetilde{u}_h - a_h|^2dx \le c\int _{ Q_1}h(x)e(\widetilde{u}_h):e(\widetilde{u}_h)dx.\) 
By (REF ) and (), there exists \(u\in L^2( Q_1)\)  such that up to a subsequence
\((\widetilde{u}_h - a_h) \chi _{ Q_1\setminus \widetilde{\omega }_h} \rightharpoonup u\) 
weakly in \(L^2( Q_1).\)  Furthermore from (REF ) and () we obtain
\(\sup \limits _{h\ge 1} \int _{ Q_1} |e(\widetilde{u}_h-a_h)|^2\,dx +\mathcal {H}^1(J_{\widetilde{u}_h}) <\infty ,\) 
and hence, by [2]} there
exist a subsequence still denoted by \(\lbrace \widetilde{u}_h-a_h\rbrace \) 
for which the set
\(E:=\lbrace y\in Q_1:\,\,\lim \limits _{h\rightarrow \infty }|\widetilde{u}_h(y) -a_h(y)|\rightarrow \infty \rbrace \) 
has finite perimeter and \(\widetilde{u}\in GSBD^2( Q_1\setminus E;\mathbb {R}^2)\)  with
\(\widetilde{u}=0\)  in \(E\)  such that
\({\begin{array}{c}\widetilde{u}_h-a_h \rightarrow \widetilde{u}\qquad \text{a.e. in $ Q_1\setminus E$}\\e(\widetilde{u}_h-a_h) \rightharpoonup e(\widetilde{u})\qquad \text{in $L^2( Q_1\setminus E;\mathbb {M}^{2\times 2}_{\rm sym}),$}\\\mathcal {H}^1(( Q_1\setminus \color {black}\partial ^* E\color {black})\cap J_{\widetilde{u}})+ \mathcal {H}^1( Q_1\cap \color {black}\partial ^* E\color {black}) =\mathcal {H}^1(J_{\widetilde{u}} \cup \partial ^*E) \le \liminf \limits _{h\rightarrow +\infty } \mathcal {H}^1(J_{\widetilde{u}_h}) =0.\end{array}}\) 
In particular, \(P(E,Q_1)=0\)  so that by the relative isoperimetric inequality either \(E=Q_1\)  or \(E=\emptyset .\)  By the definition of \(E,\)  (), the uniform \(L^2( Q_1)\) -boundedness of \(\lbrace (\widetilde{u}_h-a_h)\chi _{ Q_1\setminus \widetilde{\omega }_h}\rbrace \)  which is a consequence of (REF ) and (), and Fatou's Lemma it follows that \(|E|=0.\)  Hence, from (REF ) we get \(\widetilde{u}_h-a_h \rightarrow \widetilde{u}\) 
a.e. in \( Q_1\)  and \(e(\widetilde{u}_h-a_h) \rightharpoonup e(\widetilde{u})\)  in \(L^2( Q_1;\mathbb {M}^{2\times 2}_{\rm sym}),\)  and all relations in (REF ) hold in \( Q_1\)  and \(\widetilde{u} = u\)  a.e. in \( Q_1.\)  In particular, since \(\mathcal {H}^1(J_u)=0,\)  by the Poincaré-Korn inequality \(u\in H^1( Q_1;\mathbb {R}^2)\) . In view of the fact that our elastic energy is invariant under rigid deformations, we suppose \(a_h=0\)  for any \(h\ge 1.\)
Next we prove (). Let \(v\in H^1( Q_1;\mathbb {R}^2)\)  be such that \(\mathrm {supp}\,(u-v)\subset \subset Q_r\)  for some \(r\in (0,1).\) 
Let \(\psi \in C_c^1( Q_r;[0,1])\)  be a cut-off function with
\(\lbrace 0<\psi <1\rbrace \subset \lbrace u=v\rbrace \cap Q_{r^{\prime }}\)  and \(\mathrm {supp}\,(u-v)\subseteq \lbrace \psi \equiv 1\rbrace \subseteq Q_{r^{\prime \prime }}\)  for some \(r^{\prime \prime }<r^{\prime }<r.\) 
By (REF ) and Lemma REF  applied with \((A_h,u_h)\)  and \( Q_r\)  there exist \(\widetilde{v}_h\in GSBD^2({\rm Int}{(\Omega \cup S\cup \Sigma )};\mathbb {R}^2),\)  \(B_h\in \mathcal {A}_{m_h}\)  with \((B_h,\widetilde{v}_h\big |_{B_h})\in \mathcal {C}_{m_h},\)  \(\color {black}r_h\in (r(1-\sqrt{\delta _h}),r)\)  and a Lebesgue measurable set \(\omega _h\subset \subset Q_r\)  such that
(a1)
\(\widetilde{v}_h\in C^\infty ( Q_{r(1-\sqrt{\delta _h})}),\)  \( A_h\Delta B_h\subset \subset \color {black}Q_{r_h}\setminus Q_{r(1-\sqrt{\delta _h})}\)  and \(\mathrm {supp}\,(\widetilde{u}_h-\widetilde{v}_h)\subset \subset Q_r;\)
(a2)
\(\mathcal {H}^1(\partial B_h\setminus \partial A_h) \le c_0\,\sqrt{\delta _h}\,\mathcal {H}^1([ Q_r\setminus Q_{r(1-\sqrt{\delta _h})}]\cap \partial A_h );\)
(a3)
\(|\omega _h|\le c_0 \delta _h\,\mathcal {H}^1( Q_r\cap \partial A_h )\)  and
\(\int _{ Q_r\setminus \omega _h} |\widetilde{v}_h-\widetilde{u}_h|^2dx \le c_0\delta _h^2r^2\,\int _{ Q_r\cap A_h}|e(u_h)|^2dx;\) 
(a4)
for any \(\eta \in \mathrm {Lip}( Q_r;[0,1])\)
\(\int _{ Q_r} \eta he(\widetilde{v}_h):e(\widetilde{v}_h)dx \le & \int _{ Q_r\cap A_h} \eta he(u_h):e(u_h)dx\nonumber \\&+ d_3\,\delta _h^s\,(1+r\,\mathrm {Lip}(\eta )) \int _{ Q_r\cap A_h}|e(u_h)|^2dx,\) 


where \(\delta _h:=r^{-1/2}\mathcal {H}^1( Q_r\cap \partial A_h )^{1/2}\rightarrow 0,\)  and \(d_3\)  and \(s\)  are constants. We assume that \(h\)  is large enough so that \(r_h>r^{\prime }.\)  Set
\(v_h:=(1-\psi ) \widetilde{v}_h+ \psi v.\) 
We observe that \(\mathrm {supp}\,(u_h - v_h\big |_{B_h})\subset \subset Q_r:\)  by (a1) and the definition of \(\psi ,\)  there exists \(r_0\in (r_h,r)\)  such that \(A_h\setminus Q_{r_0} = B_h\setminus Q_{r_0}\)  and \(\widetilde{u}_h = \widetilde{v}_h=v_h\)  in \(Q_r\setminus Q_{r_0}\)  and hence, \(u_h\big |_{Q_r\cap A_h\setminus Q_{r_0}} = \widetilde{u}_h\big |_{Q_r\cap A_h\setminus Q_{r_0}} = v_h\big |_{Q_r\cap B_h\setminus Q_{r_0}}.\)  Thus, \((B_h,v_h)\)  is an admissible configuration in (REF ) and from () and the definition of deviation it follows that
\(\mathcal {F}_h(A_h,u_h; Q_1) \le \mathcal {F}_h(B_h,v_h; Q_1) +o(1),\) 
where \(o(1)\rightarrow 0\)  as \(h\rightarrow \infty .\)  We observe that
\(\mathcal {S}_h(B_h; Q_1) - \mathcal {S}_h(A_h; Q_1) \le & \mathcal {S}_h(B_h; Q_r\setminus \overline{ Q_{r(1-\sqrt{\delta _h})}}) \color {black}- \mathcal {S}_h(A_h; Q_r\setminus \overline{ Q_{r(1-\sqrt{\delta _h})}}) \\\color {black}\le & \color {black}\int _{(\partial ^*B_h\setminus \partial ^*A_h) \cap Q_r\setminus \overline{ Q_{r(1-\sqrt{\delta _h})}} } \varphi (x,\nu _{B_h}) d\mathcal {H}^1\\\color {black}& \color {black}+ 2 \int _{(Q_r\setminus \overline{ Q_{r(1-\sqrt{\delta _h})}})\cap (B_h^{(1)}\cup B_h^{(0)})\cap (\partial B_h\setminus \partial A_h) } \varphi (x,\nu _{A_h}) d\mathcal {H}^1\\\le & 2c_{2,h} \mathcal {H}^1(\partial B_h\setminus \partial A_h)\le 2c_0c_{2,h}\sqrt{\delta _h} \mathcal {H}^1([ Q_r\setminus Q_{r(1-\sqrt{\delta _h})}]\cap \partial A_h ) \\\le &\frac{2c_0\sqrt{\delta _h}}{d_5}\mathcal {S}_h(A_h; Q_1) =o(1)\) 

as \(h\rightarrow +\infty ,\)  where we used in the first inequality (a1), in the second the definition and nonnegativity of \(\mathcal {S}_h\) , in the third (REF ), in the fourth (a2) in the last again (REF ) and the definition of \(\mathcal {S}_h,\)  and finally in the equality we used (). Thus, (REF ) is rewritten as
\(\mathcal {W}_h(A_h,u_h; Q_1) \le \mathcal {W}_h(B_h,v_h; Q_1) +o(1).\) 
Note that
by (a1), (a3), (REF ), (REF ) and Fatou's Lemma, \(\widetilde{v}_h\chi _{Q_r\setminus \omega _h}\rightarrow u\)  a.e. in \( Q_r\)  and by (a3) \(\chi _{Q_r\setminus \omega _h} \rightarrow 1\)  a.e. in \(Q_r.\)  Therefore, for a.e. \(x\in Q_r\)  there exists \(h_x\ge 1\)  such that \(\chi _{Q_r\setminus \omega _h}(x)=1\)  for every \(h>h_x\)  and \(\widetilde{v}_h(x)=\widetilde{v}_h(x)\chi _{Q_r\setminus \omega _h}(x)\rightarrow u(x).\)  So
\(\text{$\widetilde{v}_h\rightarrow u$ a.e. in $Q_r.$}\) 
We claim that \(\widetilde{v}_h\rightarrow u\)  strongly in \(L_\mathrm {loc}^2( Q_r).\)  To see this we fix \(\rho \in (0,r),\)  and, since \(\delta _h\rightarrow 0\)  by (a1), there exists \(h_\rho \ge 1\)  such that \(\widetilde{v}_h\in H^1(Q_\rho )\)  for every \(h>h_\rho .\) 
From (REF ), () and (REF ) as well as the Korn-Poincaré inequality
\(\sup \limits _{h>h_\rho } \Vert \widetilde{v}_h - b_h\Vert _{H^1(Q_\rho )} <\infty \) 
for some sequence \(\lbrace b_h\rbrace \)  of rigid displacements.
On the one hand, by Rellich-Kondrachov Theorem there exist \(z\in H^1( Q_\rho ;\mathbb {R}^2)\)  and not relabelled subsequence such that
\(\widetilde{v}_h - b_h \rightarrow z\)  in \(L^2( Q_\rho ;\mathbb {R}^2)\)  and a.e. in \(Q_\rho .\)  On the other hand, by (REF ) \(b_h = \widetilde{v}_h - (\widetilde{v}_h -b_h)\)  converges to \(b:=u-z\)  a.e. in \(Q_\rho .\)  Since \(b_h\)  is a rigid displacement, so is \(b\)  and hence \(b_h\rightarrow b\)  uniformly in \(Q_\rho .\)  Therefore,
\(\limsup \limits _{h\rightarrow \infty } \Vert \widetilde{v}_h - u\Vert _{L^2( Q_\rho )}^{}\le \limsup \limits _{h\rightarrow \infty } \Vert \widetilde{v}_h -b_h - z\Vert _{L^2( Q_\rho )}^{}+\limsup \limits _{h\rightarrow \infty } \Vert b_h -b\Vert _{L^2( Q_\rho )}^{} =0,\) 
and the claim follows.
Since \(u=v\)  out of \(\lbrace \psi =1\rbrace ,\)  the claim implies
\(\widetilde{v}_h\rightarrow v\)  strongly in \(L^2(\lbrace 0<\psi <1\rbrace ),\)  and hence,
\(\lim \limits _{h\rightarrow \infty } \int _{ Q_r} |\nabla \psi \odot (v- \widetilde{v}_h)\big |_{A_h}|^2 \le &\liminf \limits _{h\rightarrow \infty } \int _{\lbrace 0<\psi <1\rbrace } |\nabla \psi \odot (v- \widetilde{v}_h)|^2 = 0,\) 

where \(X\odot Y = (X\otimes Y +Y\otimes X)/2,\) 
Thus, by the definition of \(v_h\)  and the equality
\(e(v_h) = (1-\psi )e(\widetilde{v}_h) + \psi e(v) +\nabla \psi \odot (v- \widetilde{v}_h),\) 
we estimate
\(&\int _{ Q_r}h e(v_h):e(v_h)dx\nonumber \\=& \int _{ Q_r} (1-\psi )^2h e(\widetilde{v}_h):e(\widetilde{v}_h)dx+ \int _{ Q_r}\psi ^2h e(v):e(v)dx\nonumber \\& + \int _{ Q_r} h (\nabla \psi \odot (v- \widetilde{v}_h)) :( \nabla \psi \odot (v- \widetilde{v}_h))dx\nonumber \\& + 2\int _{ Q_r} (1-\psi ) h e(\widetilde{v}_h):(\nabla \psi \odot (v- \widetilde{v}_h))dx\nonumber \\& +2\int _{ Q_r} \psi he(v):(\nabla \psi \odot (v- \widetilde{v}_h))dx\nonumber \\=& \int _{ Q_r} (1-\psi )^2h e(\widetilde{v}_h):e(\widetilde{v}_h)dx+ \int _{ Q_r}\psi ^2h e(v):e(v)dx +o(1)\nonumber \\\le &\int _{ Q_r\cap A_h} (1-\psi )^2h e(u_h):e(u_h)dx+ \int _{ Q_r}\psi ^2h e(v):e(v)dx + o(1),\) 

where in the second equality we use (), (REF ) with \(\eta \equiv 1,\)  (REF ), (REF ) and the Hölder inequality, while in the last inequality we use (REF ) with \(\eta =(1-\psi )^2\)  and (REF ).
Now (REF ), (REF ) and (REF ) imply
\(\int _{ Q_r}(2\psi -\psi ^2)h e(\widetilde{u}_h):e(\widetilde{u}_h)dx \le \int _{ Q_r}\psi ^2h e(v):e(v)dx + o(1).\) 

Since \(\lbrace h\rbrace \)  is equibounded (see (REF )) and equicontinuous, by the Arzela-Ascoli Theorem, there exist a (not relabelled) subsequence and an elasticity tensor \(C^0( Q_1;\mathbb {M}^{2\times 2}_{\rm sym})\)  such that \(h\rightarrow  uniformly in \)  Q1.\( Hence,letting \) h\( in (\ref {djksdfkj}) and using the convexity of the elastic energy and (\ref {safafaess}), we obtain\begin{equation}\int _{ Q_r}(2\psi -\psi ^2) y)e(u):e(u)dy \le \int _{ Q_r} \psi ^2 y)e(v):e(v)\,dy.\end{equation}By the choice of \) ,\( (\ref {shapatoyoqcha}) implies\begin{equation}\int _{ Q_{r^{\prime \prime }}} y)e(u):e(u)dy \le \int _{ Q_r} y)e(v):e(v)\,dy.\end{equation}Since \) r”\( is arbitrary, letting \) r”r\( we deduce that (\ref {ajsdhjss}) holds also with \) r”=r.\( Since \) supp (u-v)Qr\(, this implies (\ref {local_minimal_ekanku}).\) It remains to prove (REF ).
If we take \(v=u\)  in (REF ) and use \(0\le \psi \le 1\)  and \(\psi =1\)  in \( Q_{r^{\prime \prime }}\) 
we get
\(&\int _{ Q_{r^{\prime \prime }}}{u}:e(u)dx\le \liminf \limits _{h\rightarrow \infty } \int _{ Q_{r^{\prime \prime }}}h e(\widetilde{u}_h):e(\widetilde{u}_h)dx \nonumber \\&\le \limsup \limits _{h\rightarrow \infty } \int _{ Q_{r^{\prime \prime }}}h e(\widetilde{u}_h):e(\widetilde{u}_h)dx \le \int _{ Q_r}{u}:e(u)dx.\) 

Since \(r^{\prime \prime }\)  is arbitrary, letting \(r^{\prime \prime }\nearrow r \)  we deduce
\(\lim \limits _{h\rightarrow \infty } \int _{ Q_r}h e(\widetilde{u}_h):e(\widetilde{u}_h)dx =\int _{ Q_r}{u}:e(u)dx.\) 
Now we prove that
\(\lim \limits _{h\rightarrow \infty } \mathcal {S}_h(A_h; Q_r) =0\) 
for any \(r\in (0,1).\) 
By (), we can find \(h_r>0\)  such that
\(\color {black}\mathcal {H}^1( Q_1\cap \partial A_h )<(1-r)/5\) 
for any \(h>h_r,\)  and hence
there is no connected component of \(\partial A_h\)  intersecting both \(\partial Q_r\)  and \(\partial Q_1.\)  Also by the relative isoperimetric inequality, passing to further subsequence we suppose that either
\(\lim \limits _{h\rightarrow \infty } | Q_1\cap A_h| =0\) 
or
\(\lim \limits _{h\rightarrow \infty } | Q_1\setminus A_h| =0.\) 
First assume that (REF ) holds. Let \(E_h\subset A_h\)  be the set consisting of all connected components of \(\overline{A_h}\)  not intersecting \(\partial Q_1.\) 
Then, \((A_h\setminus E_h,u_h\big |_{A_h\setminus E_h})\)  is an admissible configuration in (REF ),
thus,
\(\mathcal {F}_h(A_h,u_h; Q_1) \le \Phi _h(A_h,u_h; Q_1) +o(1) \le \mathcal {F}_h(A_h\setminus E_h,u_h; Q_1) + o(1),\) 
where in the first inequality we use () and in the second we use the definition of \(\Phi _h.\)  Hence,
\(\begin{aligned}\mathcal {S}(A_h; Q_r)\le & \mathcal {S}(E_h;Q_1) = \mathcal {S}_h(A_h; Q_1) - \mathcal {S}_h(A_h\setminus E_h; Q_1) \\\le & \mathcal {F}_h(A_h; Q_1) - \mathcal {F}_h(A_h\setminus E_h; Q_1) \le o(1),\end{aligned}\) 
where we used in the first inequality the definition of \(E_h,\)  which entitles that \(U_r\cap \partial A_h \subset \partial E_h \) , in the equality the disjointness of \(\overline{A_h\setminus E_h}\)  and \(\overline{E_h}\)  which follows by (REF ), and in the second inequality the nonnegativity of the elastic energy and in the third (REF ). Hence, (REF ) follows.
Now assume that (REF ) holds. Let \(\delta _h:= \color {black}r^{-1/2}\sqrt{\mathcal {H}^1(Q_r\cap \partial A_h )}\rightarrow 0\)  and let \(\psi ,\)  and \(r^{\prime \prime }< r^{\prime }<r\)  and \(v_h\)  be as in (REF ) with \(v=u\) .
Fix any \(\rho \in (0,r).\)  By (), we can find \(h_{r,\rho }>0\)  such that
\(\delta _h<\min \lbrace 1-r,r-\rho \rbrace /5\)  for any \(h>h_{r,\rho }.\)  Since \(\color {black}A_h\in \mathcal {A}_{m_h},\)  no connected component of \(\partial A_h\)  intersect both \(\partial Q_r\)  and \(\partial Q_\rho .\) 
Let \(\color {black}F_h\subset Q_1\setminus A_h\)  be the union of all connected components of \(\overline{ Q_1\setminus A_h}\)  lying strictly inside \( Q_1\)  (so \(\color {black}F_h\)  is a union of “holes” and \(\color {black}\partial F_h\subset \partial A_h\) ).
Let \(\psi \in C_c^1( Q_r;[0,1])\)  be a cut-off function with \(\lbrace 0<\psi <1\rbrace \subset \lbrace u=v\rbrace \cap Q_{r^{\prime }}\)  and \(\mathrm {supp}\,(u-v)\subseteq \lbrace \psi \equiv 1\rbrace \subseteq Q_{r^{\prime \prime }}\)  for some \(r^{\prime \prime }<r^{\prime }<r.\)  Set \(A_h^{\prime }:=A_h\cup \overline{\color {black}F_h}.\)  Applying Lemma REF  with \((A_h^{\prime },\widetilde{u}_h\big |_{A_h^{\prime }}),\)  \( Q_r\)  and \(m=m_h\)  we find \(\widetilde{v}_h^{\prime }\in GSBD^2({\rm Int}{(\Omega \cup S\cup \Sigma )};\mathbb {R}^2),\)  \(B_h^{\prime }\in \mathcal {A}_{m_h}\)  with \((B_h^{\prime },\widetilde{v}_h^{\prime }\big |_{B_h})\in \mathcal {C}_{m_h},\)  \(\color {black}r_h\in (r(1-\sqrt{\delta _h}),r)\)  and a Lebesgue measurable set \(\omega _h^{\prime }\subset \subset Q_r\)  such that
(b1)
\(\widetilde{v}_h^{\prime }\in C^\infty ( Q_{r(1-\sqrt{\delta _h})}),\)  \(A_h^{\prime }\Delta B_h^{\prime }\subset \subset \color {black}Q_{r_h}\setminus Q_{r(1-\sqrt{\delta _h})}\)  and \(\mathrm {supp}\,(\widetilde{u}_h-\widetilde{v}_h^{\prime })\subset \subset Q_r;\)
(b2)
\(\mathcal {H}^1(\partial B_h^{\prime }\setminus \partial A_h^{\prime }) \le c_0\,\sqrt{\delta _h}\,\mathcal {H}^1([ Q_r\setminus Q_{r(1-\sqrt{\delta _h})}]\cap \partial A_h^{\prime } );\)
(b3)
\(|\omega _h^{\prime }|\le c_0 \delta _h\,\mathcal {H}^1( Q_r\cap \partial A_h^{\prime })\)  and
\(\int _{ Q_r\setminus \omega _h^{\prime }} |\widetilde{v}_h^{\prime }-\widetilde{u}_h|^2dx \le c_0\delta _h^2r^2\,\int _{ Q_r\cap A_h^{\prime }}|e(u_h)|^2dx;\) 
(b4)
for any \(\eta \in \mathrm {Lip}( Q_r;[0,1])\)
\(\int _{ Q_r} \eta {\widetilde{v}_h^{\prime }}:e(\widetilde{v}_h^{\prime })dx \le & \int _{ Q_r\cap A_h} \eta {u_h}:e(u_h)dx\nonumber \\&+ d_3\,\delta _h^s\,(1+r\mathrm {Lip}(\eta )) \int _{ Q_r\cap A_h}|e(u_h)|^2dx,\) 


where \(d_3\)  and \(s\)  are constants.
Set
\(v_h^{\prime }:=(1-\psi ) \widetilde{v}_h^{\prime }+ \psi u.\) 
By the definition of \(A_h^{\prime } \)  and (b1) \((B_h^{\prime },v_h^{\prime }\big |_{B_h^{\prime }})\)  is an admissible configuration for \(\Phi _h(A_h,u_h; Q_1)\)  in (REF ). Thus from () and (REF )
\(\mathcal {F}_h(A_h,u_h; Q_1) \le \mathcal {F}_h(B_h^{\prime },v_h^{\prime }\big |_{B_h^{\prime }}; Q_1) +o(1).\) 
Now as in the proof of (REF )
\(&\mathcal {W}_h(B_h^{\prime },v_h^{\prime }\big |_{B_h^{\prime }}; Q_1) - \mathcal {W}_h(A_h,u_h; Q_1)\nonumber \\\le &\int _{ Q_r}\psi ^2h e(u):e(u)dx -\int _{ Q_r}(2\psi -\psi ^2)h e(\widetilde{u}_h):e(\widetilde{u}_h)dx + o(1) \nonumber \\\le & \int _{ Q_r} h e(u):e(u)dx - \int _{ Q_{r^{\prime \prime }}} h e(\widetilde{u}_h):e(\widetilde{u}_h)dx + o(1).\) 

Moreover,
\(& \mathcal {S}_h(B_h^{\prime }; Q_1) - \mathcal {S}_h(A_h; Q_1)= \color {black}\Big (\mathcal {S}_h(B_h^{\prime }; Q_1) - \mathcal {S}_h(A_h^{\prime }; Q_1) \Big ) + \Big (\mathcal {S}_h(A_h^{\prime }; Q_1) - \mathcal {S}_h(A_h; Q_1) \Big ) \color {black}\nonumber \\\le & \mathcal {S}_h(B_h^{\prime }; Q_r\setminus Q_{r(1-\sqrt{\delta _h})}) - \mathcal {S}_h(A_h; Q_\rho ) \le 2c_{2,h} \mathcal {H}^1(\partial B_h^{\prime }\setminus \partial A_h^{\prime })- \mathcal {S}_h(A_h; Q_\rho ) \nonumber \\\le & 2c_0c_{2,h}\sqrt{\delta _h} \mathcal {H}^1([ Q_r\setminus Q_{r(1-\sqrt{\delta _h})}]\cap \partial A_h) - \mathcal {S}_h(A_h; Q_\rho ) \nonumber \\\le & \frac{2c_0\sqrt{\delta _h}}{d_5}\mathcal {S}_h(A_h; Q_1) - \mathcal {S}_h(A_h; Q_\rho ) = o(1) - \mathcal {S}_h(A_h; Q_\rho ),\) 

where we used in the first inequality (b1) and the definition of \(A_h^{\prime },\)  in the second and in the last inequalities the definition of \(\mathcal {S}_h,\)  (REF ) and (REF ), in the third inequality (b2), and in the last equality () and that \(\delta _h\rightarrow 0\)  by ().
Hence, (REF ), (REF ) and (REF ) imply
\(\mathcal {S}_h(A_h; Q_\rho ) + \int _{ Q_{r^{\prime \prime }}} h e(\widetilde{u}_h):e(\widetilde{u}_h)dx \le \int _{ Q_r} h e(u):e(u)dx + o(1).\) 

Thus, letting \(h\rightarrow \infty \)  and using (REF ) we get
\(\limsup \limits _{h\rightarrow \infty } \,\mathcal {S}_h(A_h; Q_\rho ) + \int _{ Q_{r^{\prime \prime }}} {u}:e(u)dx \le \int _{ Q_r} {u}:e(u)dx.\) 
Now letting \(r^{\prime \prime }\rightarrow r\)  we get
\(\limsup \limits _{h\rightarrow \infty } \mathcal {S}_h(A_h; Q_\rho ) =0.\) 
Observe that the function \(B\mapsto \mathcal {S}_h(A_h;B)\)  defined for Borel sets \(B\subset Q_1\)  extends to a bounded nonnegative Radon measure \(\mu _h\)  in \( Q_1\) . Since (REF ) holds for any \(\rho \in (0,r),\)  \(\mu _h\)  converges to 0 in the weak* sense, and thus (REF ) follows.
Recall that by [3]}
if the elasticity tensor \( is constant,then for any \) (0,2)\( there exists\) c:=c(c3,c4)>0\( such thatfor every local minimizer \) (,u)C\(of \) F(;O),\( \) u\( is analytic in \) O\(and for any square\)  QR(x)O\( and \) r(0,R),\(\begin{equation}\int _{ Q_r(x)}{u}:e(u)\,dx \le c_\gamma \, \Big (\frac{r}{R}\Big )^{2-\gamma }\int _{ Q_R(x)} {u}:e(u)\,dx.\end{equation}Given \) (0,1)\( let\begin{equation*}\tau _0=\tau _0(\gamma ,c_3,c_4):=\min \lbrace 1,\tfrac{1}{2} c_{\gamma }^{-\frac{1}{4-2\gamma }}\rbrace ,\end{equation*}where \) c\( is the constant appearing in (\ref {c_gamma}).Using Proposition \ref {prop:conti_prop_3.4} and repeating similar arguments of \cite {ChC:2019_arxiv,CFI:2018poincare} we get the following decay property of the functional \) F\(.\)
Proposition 3.4 
For any \(\tau \in (0,\tau _0)\)  there exist
\(\varsigma = \varsigma (\tau )\in (0,1)\)  and \(\vartheta :=\vartheta (\tau )\in (0,1)\)  with the following property: If there exist \(m\in \mathbb {N}\cup \lbrace \infty \rbrace ,\)  \((A,u)\in \mathcal {C}_m\)  and a square \(Q_\rho (x)\subset \subset \Omega \)  such that
\(\mathcal {H}^1( Q_\rho (x)\cap \partial A) \le 2\varsigma \rho \quad \text{and}\quad \mathcal {F}(A,u; Q_\rho (x)) \le (1+\vartheta )\Phi (A,u; Q_\rho (x)),\) 
then
\(\mathcal {F}(A,u; Q_{\tau \rho }(x)) \le \tau ^{2-\gamma }\mathcal {F}(A,u; Q_\rho (x)).\) 


We argue by contradiction. Assume that there exists \(\tau \in (0,\tau _0)\)  such that for all \(\varsigma ,\vartheta \in (0,1)\)  we can find \(m:=m(\varsigma ,\vartheta )\in \mathbb {N}\cup \lbrace \infty \rbrace ,\)  \((A,u):=(A(\varsigma ,\vartheta ),u(\varsigma ,\vartheta ))\in \mathcal {C}_m\)  and \(Q_\rho (x)\subset \subset \Omega \)  with \(\rho :=\rho (\varsigma ,\vartheta )\)  and \(x:=x(\varsigma ,\vartheta )\)  satisfying
\(\mathcal {H}^1( Q_{\rho }(x)\cap \partial A) \le 2\varsigma \rho \quad \text{and} \quad \mathcal {F}(A,u_; Q_{\rho }(x)) \le (1 + \vartheta )\Phi (A,u; Q_{\rho }(x)),\) 

but
\(\mathcal {F}(A,u; Q_{\tau \rho }(x))>\tau ^{2-\gamma } \mathcal {F}(A,u; Q_{\rho }(x)).\) 
Let us choose any positive real numbers \(\varsigma _h,\vartheta _h\rightarrow 0,\)  and denote for simplicity \(m_h:=m(\varsigma _h,\vartheta _h),\)  \((A_h,u_h)= (A(\varsigma _h,\vartheta _h),u(\varsigma _h,\vartheta _h)),\)  \(\rho _h:=\rho (\varsigma _h,\vartheta _h),\)  \(x_h=x(\varsigma _h,\vartheta _h).\)  By (REF ) and (REF ),
\(& \mathcal {H}^1( Q_{\rho _h}(x_h)\cap \partial A_h) \le 2\varsigma _h\rho _h, \\& \mathcal {F}(A_h,u_h; Q_{\rho _h}(x_h)) \le (1 + \vartheta _h)\Phi (A_h,u_h; Q_{\rho _h}(x_h)), \) 

but
\(\mathcal {F}(A_h,u_h; Q_{\tau \rho _h}(x_h))>\tau ^{2-\gamma } \mathcal {F}(A_h,u_h; Q_{\rho _h}(x_h)) \) 
for any \(h.\)  Note that \(\mathcal {F}(A_h,u_h; Q_{\rho _h}(x_h))>0.\)  Let us define the rescaled energy \(\mathcal {F}_h(\cdot ; Q_1):\mathcal {C}_{m_h}\rightarrow \mathbb {R}\)  as in (REF ) with
\(\varphi _h(y,\nu ):= \frac{\rho _h\varphi (x_h+\rho _hy,\nu )}{ \mathcal {F}(A_h,u_h; Q_{\rho _h}(x_h))}\) 
in place of \(\varphi (y,\nu )\)  and
\(h(y):=x_h+\rho _hy)\) 
in place of \(y)\) , for \(y\in Q_1\) .
We notice that
\(\mathcal {F}_h(E_h,v_h; Q_1)=1\) 
for
\(E_h:= \sigma _{x_h,\rho _h}(A_h)\) 
(see definition of blow-up map \(\sigma _{x,r}\)  at (REF )) and
\(\color {black}v_h(y):=\frac{u_h(x_h+\rho _hy)}{\sqrt{\mathcal {F}(A_h,u_h;B_{\rho _h}(x_h))}}.\) 
By (REF ) we obtain
\(\mathcal {H}^1( Q_1\cap \partial E_h) <2\varsigma _h\) 
while () and (REF ) entails
\(\Psi _h(E_h,v_h; Q_1) \le \vartheta _h \Phi _h(E_h,v_h; Q_1) \le \vartheta _h \mathcal {F}_h(E_h,v_h; Q_1)=\vartheta _h,\) 
where \(\Phi _h\)  and \(\Psi _h\)  are defined as in (REF ) and (REF ) (again with \(\varphi _h\)  and \(h\)  in places of \(\varphi \)  and \( \)  respectively). By (REF ) \(\lbrace h\rbrace \)  is equibounded. Since \(\Omega \)  is bounded, there exists \(x_0\in \overline{\Omega }\)  such that, up to extracting a subsequence, \(x_h\rightarrow x_0\)  as \(h\rightarrow +\infty .\)  As \(\rho _h\rightarrow 0,\)  one has \(x_h+ \rho _h y\rightarrow x_0\)  for every \(y\in \overline{ Q_1}.\)  Thus \(\lbrace h\rbrace \)  is also equicontinuous and \(h\rightarrow 0:=x_0)\)  uniformly in \( \overline{ Q_1}.\) 
In view of (REF ), () and (REF ), we can apply Proposition REF  to find \(u\in H^1( Q_1;\mathbb {R}^2),\)  vectors \(\xi _h\in (0,1)^2\) , and infinitesimal rigid displacements \(a_h\)  such that, up to a subsequence,
\(w_h:=v_h\chi _{ Q_1\cap E_h}^{} + \xi _h\chi _{ Q_1\setminus E_h}^{} - a_h \rightarrow v\) 
pointwise a.e. in \( Q_1\) , \(e(w_h) \rightharpoonup e(v)\)  in \(L^2( Q_1)\)  as \(h\rightarrow +\infty ,\) 
and
\(\color {black}\lim \limits _{h\rightarrow +\infty } \mathcal {F}_h(E_h,w_h; Q_r) = \lim \limits _{h\rightarrow +\infty } \mathcal {F}_h(E_h,v_h; Q_r) =\int _{ Q_r} 0(x)e(v):e(v)dx\) 
for any \(r\in (0,1].\) 
In particular, from (REF ) and (REF ) it follows that
\(\int _{ Q_\tau } 0(x)e(v):e(v)dx&=\lim \limits _{h\rightarrow +\infty }\mathcal {F}(E_h,v_h; Q_{\tau })\\&\ge \lim \limits _{h\rightarrow +\infty }\tau ^{2-\gamma } \mathcal {F}(E_h,v_h; Q_{1}) = \tau ^{2-\gamma } \int _{ Q_1} 0(x)e(v):e(v)dx.\) 

Since 0 is constant, applying () with \(r:=\tau \)  and \(R:=1\)  we get
\(c_\gamma \tau ^{2-\gamma }\int _{ Q_1} 0(x)e(v):e(v)dx &\ge \int _{ Q_{\tau }} 0(x)e(v):e(v)dx\\&\ge \tau ^{\gamma -2}\int _{ Q_1} 0(x)e(v):e(v)dx.\) 

Now recalling that \(\mathcal {F}_h(E_h,v_h; Q_1)=1,\)  by (REF ) we get \(\int _{ Q_1} 0(x)e(v):e(v)dx =1,\)  thus, \(\tau ^{2-\gamma }\ge c_\gamma ^{-1/2}>\tau _0^{2-\gamma },\)  a contradiction.
By employing the arguments of [4]} and using Proposition REF 
we establish the following lower bound for \(\mathcal {F}\) .
Proposition 3.5 
Given \(\tau \in (0,\tau _0)\) , let \(\varsigma :=\varsigma (\tau )\in (0,1)\)  and \(\vartheta :=\vartheta (\tau )\in (0,1)\)  be as in Proposition REF . Let \((A,u)\in \mathcal {C}_m\)  be a \((\Lambda ,m)\) -minimizer of \(\mathcal {F}\)  in \( Q_{r_0}(x_0)\)  for some \(\color {black}m\in \mathbb {N}\cup \lbrace \infty \rbrace \)  and \(r_0>0,\)  and let
\(J_A^*:=\lbrace y\in Q_{r_0}(x_0) \cap \partial A:\,\, \theta _*(\partial A,y)>0\rbrace .\) 
Then
\(\mathcal {F}(A,u; Q_\rho (x)) \ge 2c_1\varsigma \rho .\) 
for every \(\color {black}x\in \overline{J_A^*}\)  and for every square \( Q_\rho (x)\subset Q_{r_0}(x_0)\)  with \(\rho \in (0,R_0),\)  where
\(R_0:=R_0(r_0,\Lambda ,c_1,\tau ):=\min \Big \lbrace r_0,\frac{\sqrt{\pi }\,c_1\vartheta }{\Lambda (2+\vartheta )}\Big \rbrace .\) 


Fix \(m\in \mathbb {N}\cup \lbrace \infty \rbrace .\)  Note that for any \((C,w),(D,v)\in \mathcal {C}_m\)  and \(O\subset \Omega \)  with \(C\Delta D\subset \subset O\)
\(\sqrt{4\pi }\,|C\Delta D|^{1/2} \le & \mathcal {H}^1(\partial ^*(C\Delta D)) \le \mathcal {H}^1(O\cap \partial ^* C) + \mathcal {H}^1(O\cap \partial ^* D)\nonumber \\\le &\frac{\mathcal {S}(C,O) + \mathcal {S}(D,O)}{c_1} \le \frac{\mathcal {F}(C,w;O) + \mathcal {F}(D,v;O)}{c_1},\) 

where in the first inequality we used the isoperimetric inequality, in the second \(\partial ^*(C\Delta D) \subset O\cap (\partial ^*C\cup \partial ^*D),\)  in the third (REF ) and the definition of \(\mathcal {S}(\cdot ;O)\)  and in the last the nonnegativity of \(\mathcal {W}(\cdot ;O).\) 
Thus, from the \((\Lambda ,m)\) -minimality of \((A,u)\)  in \( Q_{r_0}(x_0)\)  we deduce that
\(\!\mathcal {F}(A,u; Q_r(x)) \le & \mathcal {F}(B,v; Q_r(x)) + \Lambda |A\Delta B|^\frac{1}{2}|A\Delta B|^\frac{1}{2}\nonumber \\\le & \mathcal {F}(B,v; Q_r(x)) + \frac{\Lambda r}{\sqrt{\pi }\,c_1}\Big (\mathcal {F}(A,u; Q_r(x)) + \mathcal {F}(B,v; Q_r(x))\Big )\) 

for any \( Q_\rho (x)\subset Q_{r_0}(x_0)\)  and \((B,v)\in \mathcal {C}_m\)  with \(A\Delta B\subset \subset Q_r(x)\)  and \(\mathrm {supp}\,(u-v)\subset \subset Q_r(x)\) , where in the last inequality we used (REF ) and the inequality \(|A\Delta B| \le | Q_r|=4r^2.\) 
Let \(r>0\)  be small enough so that \(\frac{\Lambda r}{\sqrt{\pi }\,c_1} \le \frac{\vartheta }{2+\vartheta },\)  where \(\vartheta :=\vartheta (\tau )\in (0,1)\)  is given by Proposition REF . From (REF ) we obtain
\(\mathcal {F}(A,u; Q_r(x)) \le (1+ \vartheta )\mathcal {F}(B,v; Q_r(x)),\) 
which by the arbitrariness of \((B,v)\)  is equivalent to
\(\mathcal {F}(A,u; Q_r(x)) \le (1+ \vartheta )\Phi (A,u; Q_r(x)).\) 
We are ready to prove (REF ).
Let \(x\in J_A^*.\)  For simplicity we suppose that \(x=0.\)  Assume by contradiction that for such \(m\in \mathbb {N}\cup \lbrace \infty \rbrace ,\)  \((A,u)\in \mathcal {C}_m\)  and for some \( Q_\rho \subset \subset Q_{r_0}(x_0)\)  with \(\rho \in (0,R_0)\)  we have
\(\mathcal {F}(A,u; Q_\rho )< 2c_1\varsigma \rho .\) 
Then by the nonnegativity of the elastic energy and
(REF ),
\(2c_1\varsigma \rho > \mathcal {F}(A,u; Q_\rho ) \ge \int _{ Q_\rho \cap \partial A} \varphi (x,\nu _A)d\mathcal {H}^1\ge c_1\mathcal {H}^1( Q_\rho \cap \partial A \color {black})\) 
so that
\(\mathcal {H}^1( Q_\rho \cap \partial A\color {black})< 2\varsigma \rho .\) 
By (REF ) and (REF ) we can apply Proposition REF  and obtain that
\(\mathcal {F}(A,u; Q_{\tau \rho }) \le \tau ^{2-\gamma }\mathcal {F}(A,u; Q_\rho ) \le 2c_1\varsigma \tau ^{2-\gamma } \rho .\) 
Hence,
\(\mathcal {H}^1( Q_{\tau \rho }\cap \partial A) \le 2\varsigma \tau ^{2-\gamma } \rho < 2\varsigma \tau \rho ,\) 
where we used \(\gamma ,\tau \in (0,1)\) ,
and by induction
\(\mathcal {H}^1( Q_{\tau ^n\rho }\cap \partial A) \le 2\varsigma \tau ^{(2-\gamma )n} \rho <2\varsigma \tau ^n \rho ,\quad n\in \mathbb {N}.\) 
However, by the choice of \(x\)
\(\color {black}0< \theta _*(\partial A,x)= \color {black}\liminf \limits _{n\rightarrow +\infty }\frac{\mathcal {H}^1( Q_{\tau ^n\rho }\cap \partial A)}{2\tau ^n\rho }\le \lim \limits _{n\rightarrow +\infty }\frac{2c_1\varsigma \tau ^{(1-\gamma )n}}{2c_1} = 0,\) 
a contradiction. This contradiction implies (REF ) for \(x\in J_A^*.\)  Note that the map \(\mathcal {F}(A,u;\cdot ),\)  defined for open sets \(O\subset \subset Q_{r_0}(x_0)\)  extends to a positive Borel measure in \( Q_{r_0}(x_0),\)  hence, (REF ) is valid also for all \(x\in Q_{r_0}(x_0)\cap \overline{J_A^*}\) .
Now we are ready to prove (REF ) and (REF ).
Let \(m\in \mathbb {N}\cup \lbrace \infty \rbrace \)  and \((A,u)\)  be a \((\Lambda ,m)\) -minimizer of \(\mathcal {F}(\cdot ,\cdot ;\Omega ).\)  We begin by establishing (REF ). Let \(x\in \Omega \) , \(r\in (0,1,\mathrm {dist}(x,\partial \Omega ))\) , and \( Q_r:= Q_r(x)\) . Since \(E:=(A\setminus \overline{ Q_r})\cup \partial Q_r \in \mathcal {A}_m,\)  by the \((\Lambda ,m)\) -minimality of \((A,u)\)
\(\mathcal {F}(A,u; Q_r) \le \mathcal {F}(E,u; Q_r) + \Lambda | Q_r|.\) 
Hence, by the nonnegativity \(\mathcal {W}(A\cap Q_r,u; Q_r)\)
\(\int _{ Q_r\cap \partial A} \varphi (x,\nu _{A})d\mathcal {H}^1 \le 2\int _{\partial Q_r} \varphi (x,\nu _{ Q_r})d\mathcal {H}^1 +4 \Lambda r^2\) 
and hence (REF ) entails (REF ).
In particular, since
\(E\Delta A\subset \subset Q_\rho \)  for every \(\rho \in (r,\mathrm {dist}(x,\partial \Omega )),\) 
we also have
\(\mathcal {F}(A,u; Q_\rho ) \le & \mathcal {F}(E,u; Q_\rho ) +\Lambda | Q_r| = \mathcal {F}(E,u; Q_\rho \setminus \overline{ Q_r})+\mathcal {S}(E,u;\overline{ Q_r}) +4\Lambda r^2\\\le & \mathcal {F}(E,u; Q_\rho \setminus \overline{ Q_r}) + 2\int _{\partial Q_r} \varphi (x,\nu _{ Q_r})d\mathcal {H}^1 + 4\Lambda r^2\\\le & \mathcal {F}(E,u; Q_\rho \setminus \overline{ Q_r}) + 16c_2r + 4\Lambda r^2\) 

and hence, letting \(\rho \searrow r\)  and using \(r\le 1\)  we get
\(\mathcal {F}(A,u;\overline{ Q_r}) \le (16c_2 + 4\Lambda )r.\) 
Now assuming that \(x\)  belongs to the closure of the set \(\lbrace y\in \Omega \cap \partial A:\,\theta _*(\partial A,y)>0\rbrace ,\)  we prove (REF ).
For \(\tau _o:=\tau _0/2,\)  let \(\varsigma _o=\varsigma (\tau _o)\in (0,1)\)  and \(R_o=R_0(1,\Lambda ,c_1,\tau _o)>0\)  be as in Proposition REF . Then by (REF ),
\(\mathcal {F}(A,u; Q_{\kappa r}) \ge 2c_1\varsigma _o \kappa r\) 
for \(\kappa \in (0,1]\)  and for any square \( Q_r\subset \Omega \)  with \(r\in (0,R_o).\)  We consider \(\varsigma _*:=\varsigma (\tau _*),\)  \(\vartheta _*:=\vartheta (\tau ^*)\) , and \(R_*:=\min \lbrace R(1,\Lambda ,c_1,\tau _*),R_o\rbrace \)  as given by Proposition REF  for
\(\tau _*:=\min \lbrace \frac{\tau _0}{2},\big (\frac{c_1\varsigma _o}{16c_2+4\Lambda }\big )^{\frac{1}{1-\gamma }}\rbrace \) . By contradiction, if \(\mathcal {H}^1( Q_r\cap \partial A)<\varsigma _*r,\) 
then by applying (REF ) with \(\kappa =\tau _*\)  we obtain
\(\mathcal {F}(A,u; Q_r) \le (1 + \vartheta _*) \Phi (A,u; Q_r).\) 
Then by Proposition REF ,
\(\mathcal {F}(A,u; Q_{\tau _* r}) \le \tau _*^{2-\gamma } \mathcal {F}(A,u; Q_r)\) 
so that by (REF ) and (REF )
\(\tau _*^{1-\gamma } \ge \frac{2c_1\varsigma _o}{16c_2 + 4\Lambda },\) 
which is a contradiction.

Compactness and lower-semicontinuity properties
For the convenience of the reader, we divide the prove into several propositions.
We start by showing the compactness of free crystal regions of the sequence of constrained minimizers \(\lbrace (A_m,u_m)\rbrace \) .
Proposition 4.1 
Assume that either \(\mathtt {v}\in (0,|\Omega |)\)  or \(S=\emptyset .\)  There exist \(m_h\nearrow +\infty ,\)  \((A_{m_h},u_{m_h})\in \mathcal {C}_{m_h}\)  and \(A\in \color {black}\widetilde{\mathcal {A}}\color {black}\)  such that
(a)
for any \(h\in \mathbb {N}\)  \((A_{m_h},u_{m_h})\)  is a minimizer of \(\mathcal {F}\)  in \(\mathcal {C}_{m_h}\)  with \(|A|=\mathtt {v}\)  such that \(\partial A_{m_h}\)  does not contain isolated points;
(b)
\(\mathrm {sdist}(\cdot ,\partial A_{m_h}) \rightarrow \mathrm {sdist}(\cdot ,\partial A)\)  locally uniformly in \(\mathbb {R}^2\)  as \(h\rightarrow \infty ;\)
(c)
for any \(x\in \Omega \cap \partial A\)  and \(r\in (0,\min \lbrace R_*,\mathrm {dist}(x,\partial \Omega )\rbrace )\)
\(\color {black}\frac{c_1\varsigma _*}{8\pi c_2} \le \frac{\mathcal {H}^1( Q_r(x) \cap \partial A )}{2r} \le \frac{2\pi c_2}{c_1\varsigma _*},\) 
where \(\varsigma _*:=\varsigma _*(c_3,c_4)\in (0,1)\)  and \(R_*:=R_*(c_1,c_2,c_3,c_4)>0\)  are given in Theorem REF .


By [5]} there exists a minimizer \((A_m,u_m)\in \mathcal {C}_m\)  for every \(m\in \mathbb {N}\) . Without loss of generality we assume that \(\partial A_m\)  does not contain isolated points. In fact, if \(\partial A_m\)  has a isolated point \(x\)  in \(A_m^{(0)},\)  then \(A_m\setminus \lbrace x\rbrace \in \mathcal {A}_m\)  and \(\mathcal {F}(A_m,u_m) = \mathcal {F}(A_m\setminus \lbrace x\rbrace ,u_m).\)  Analogously, if \(\partial A_m\)  has an isolated point in \(A_m^{(1)},\)  then there exists \(r>0\)  such that \(B_r(x)\cap \partial A_m=\lbrace x\rbrace \)  (and \(B_r(x)\subset A_m\cup \lbrace x\rbrace  \in \mathcal {C}_m\) ). In view of Lemma REF  the function \(u_m,\)  arbitrarily extended to \(x\)  belongs to \(H_\mathrm {loc}^1(B_r(x)),\)  hence, the configuration \((A_m\cup \lbrace x\rbrace ,u_m)\in \mathcal {C}_m\)  and satisfies \(\mathcal {F}(A_m,u_m)=\mathcal {F}(A_m\cup \lbrace x\rbrace ,u_m).\)
In view of Remark REF  \((A_m,u_m-u_0)\)  is a \((\lambda _0,m)\) -minimizer of \(\mathcal {F}(\cdot ,\cdot ;\Omega ).\)  Moreover, since \(\partial A_m\)  does not contain isolated points \(\theta _*(\partial A_m,x)>0\)  for any \(x\in \partial A_m,\)  hence by
Theorem REF  the density estimates (REF ) and (REF ) hold for all \(x\in \Omega \cap \partial A_m.\)
By [5]}, there exist \(A\subset \Omega \)  and a
subsequence \(\lbrace (A_{m_h},u_{m_h})\rbrace \)  such that
\(\mathrm {sdist}(\cdot ,\partial A_{m_h})\rightarrow \mathrm {sdist}(\cdot ,\partial A)\)  as \(h\rightarrow \infty .\) 
Consider the sequence \(\mu _h:=\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits \partial A_{m_h}\)  of positive Radon measures. By Theorem  REF
\(\frac{\varsigma _*}{2} \le \frac{\mu _h( Q_r(x))}{2r} \le \frac{2\pi c_2}{c_1}\) 
for every \(x\in \Omega \cap \partial A_{m_h}\)  and \( Q_r(x)\subset \subset \Omega \)  with \(r\in (0,R_*).\)  By (REF ), (REF ) and (REF ),
\(\begin{aligned}\mu _h(\mathbb {R}^2) = \mathcal {H}^1(\partial A_{m_h}) \le &\mathcal {H}^1(\partial \Omega )+ \frac{\mathcal {F}(A_{m_h},u_{m_h}) + 2c_2\mathcal {H}^1(\Sigma )}{c_1} \\\le &\mathcal {H}^1(\partial \Omega )+ \frac{\mathcal {F}(A_1,u_1) + 2c_2\mathcal {H}^1(\Sigma ) }{c_1},\end{aligned}\) 
hence, by compactness, there exist a not relabelled subsequence and a positive Radon measure \(\mu \)  in \(\mathbb {R}^2\)  such that
\(\mu _h\rightharpoonup ^*\mu \)  as \(h\rightarrow \infty .\) 
We claim that
\(\overline{\Omega \cap \partial A} \subseteq \mathrm {supp}\,\mu \subseteq \partial A.\) 
Indeed, let \(x\in \Omega \cap \partial A\)  and \(r\in (0,\min \lbrace \mathrm {dist}(x,\partial \Omega ),R_*\rbrace ).\) 
By the \(\mathrm {sdist}\) -convergence, there exists \(x_h\in Q_r(x)\cap \partial A_{m_h}\)  with \(x_h\rightarrow x\)  such that \( Q_{r/2}(x_h)\subset Q_r(x)\)  and hence, by the weak* convergence and (REF ),
\(\mu (\overline{ Q_r(x)}) \ge \limsup \limits _{h\rightarrow \infty }\mu _h(\overline{ Q_r(x)})\ge \limsup \limits _{h\rightarrow \infty }\mu _h( Q_{r/2}(x_h)) \ge \varsigma _* r.\) 
This implies \(x\in \mathrm {supp}\,\mu .\) 
Conversely,
if, by contradiction, there exists \(x\in \mathrm {supp}\,\mu \setminus \partial A,\) 
then we can find \(r>0\)  such that \( Q_r(x)\cap \partial A=\emptyset .\)  From the \(\mathrm {sdist}\) -convergence
it follows that \( Q_{r/2}(x)\cap \partial A_{m_h}=\emptyset \)  for \(h\)  large enough, and hence,
\(0<\mu ( Q_{r/2}(x)) \le \liminf \limits _{h\rightarrow \infty } \mu _h( Q_{r/2}(x)) =0,\) 
which is a contradiction.
From (REF ) it follows that
\(\frac{\varsigma _*}{2} \le \frac{\mu ( Q_r(x))}{2r} \le \frac{2\pi c_2}{c_1}\) 
for any \(x\in \Omega \cap \mathrm {supp}\,\mu \)  any \(r\in (0,R_*)\)  with \( Q_r(x)\subset \subset \Omega \) . Indeed, let \(x\in \Omega \cap \mathrm {supp}\,\mu \)  and let \(R(x):=\min \lbrace R_*,\mathrm {dist}(x,\partial \Omega ).\)  Then by the weak* convergence \(\mu _h(Q_r(x))\rightarrow \mu (Q_r(x))=0\)  for a.e. \(r\in (0,R(x)).\)  In particular, (REF ) holds for a.e. \(r\in (0,R(x)).\)  Since \(\mu \)  is a Radon measure, (REF ) extends to all \(r\in (0,R(x))\)  by the left-continuity of the map \(r\mapsto \mu (Q_r(x))\) .
From (REF ) and [7]} it follows that
\(\varsigma _*\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Omega \cap \mathrm {supp}\,\mu ) \le \mu \mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits \Omega \le \frac{4\pi c_2}{c_1}\,\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Omega \cap \mathrm {supp}\,\mu ).\) 
Thus, \(\color {black}\mu \mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits \Omega \)  is absolutely continuous with respect to \(\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Omega \cap \mathrm {supp}\,\mu )\)  and \(\mathcal {H}^1(\mathrm {supp}\,\mu )<\infty .\) 
By (REF ),
\(\mathcal {H}^1(\partial A) \le \mathcal {H}^1(\Omega \cap \partial A) +\mathcal {H}^1(\partial \Omega \cap \partial A) \le \frac{1}{\varsigma _*}\,\mu (\Omega ) +\mathcal {H}^1(\partial \Omega ) <\infty .\) 
In view of (REF ) for any \(x\in \Omega \cap \partial A\)
\(\varsigma _* <\limsup \limits _{r\rightarrow 0} \frac{\mu (Q_r(x))}{r} \le \frac{2\pi c_2}{c_1\varsigma _*}\, \liminf \limits _{r\rightarrow 0} \frac{\mu (Q_r(x))}{r} <\frac{4\pi ^2c_2^2}{c_1^2\varsigma _*}.\) 
Therefore by
Finally let us prove (REF ). Fix any \(x\in \Omega \cap \partial A\)  and let \(R(x):=\min \lbrace R_*,\mathrm {dist}(x,\partial \Omega )\rbrace .\)  Then by (REF )
\(\frac{\varsigma _* \mathcal {H}^1(Q_r(x)\cap \partial A)}{2r} \le \frac{\mu (Q_r(x))}{2r} \le \frac{4\pi c_2}{c_1}\,\frac{\mathcal {H}^1(Q_r(x)\cap \partial A)}{2r}.\) 
This and (REF ) imply
\(\frac{\varsigma _* \mathcal {H}^1(Q_r(x)\cap \partial A)}{2r} \le \frac{2\pi c_2}{c_1}\quad \text{and}\quad \frac{\varsigma _*}{2}\le \frac{4\pi c_2}{c_1}\,\frac{\mathcal {H}^1(Q_r(x)\cap \partial A)}{2r},\) 
and hence, (REF ) follows.
Corollary 4.2 
Let \(\lbrace A_{m_h}\rbrace \)  and \(A\)  be as in Proposition REF . Then \(A_{m_h}\rightarrow A\)  in \(L^1(\mathbb {R}^2)\)  as \(h\rightarrow \infty .\)

Since \(\mathcal {H}^1(\partial A)<\infty \)  and \(A_{m_h} \overset{K}{\rightarrow } \overline{A}\)  as \(h\rightarrow \infty ,\)  one has \(\chi _{A_{m_h}} (x) \rightarrow \chi _A(x)\)  as \(h\rightarrow \infty \)  for a.e. \(x\in \mathbb {R}^2.\)  Now Corollary REF  follows from the Dominated Convergence Theorem.
The following result generalizes [8]} since it applies to set \(\Gamma \)  a priori not connected (even not necessarily \(\mathcal {H}^1\) -rectifiable), but satisfying uniform density estimates. Recall that we denote by \(\Gamma ^r\)  and \(\Gamma ^u\)  the \(\mathcal {H}^1\) -rectifiable and purely unrectifiable parts of a Borel 1-set \(\Gamma .\)
Proposition 4.3 
Let \(\Gamma \subset \mathbb {R}^2\)  be a Borel set such that \(\mathcal {H}^1(\Gamma )<+\infty \)  and for some \(r_0,c,C>0\)  and for all \(x\in \Gamma \)
\(c\le \frac{\mathcal {H}^1( Q_r(x))}{2r} \le C,\qquad r\in (0,r_0).\) 
Then for any \(R>0\)  and a.e. \(x\in \color {black}\Gamma ^r\)  one has
\(\overline{ Q_{R,\nu _\Gamma (x)} (x)} \cap \sigma _{x,\rho }(\Gamma ) \overset{K}{\rightarrow } \overline{ Q_{R,\nu _\Gamma (x)} (x) } \cap T_x\) 
and
\(\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits ( \sigma _{x,\rho }(\Gamma ) ) \overset{*}{\rightharpoonup } \mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits T_x\) 
as \(\rho \rightarrow 0,\)  where \(\sigma _{x,r}\)  denotes the blow-up map defined in (REF ) and \(T_x\)  is the generalized tangent line to \(\Gamma \)  at \(x.\)  Moreover, for any \(\mathcal {H}^1\) -measurable \(\Gamma ^{\prime }\subset \Gamma \)  and \(\mathcal {H}^1\) -a.e. \(x\in [\Gamma ^{\prime }]^r\)  the relations (REF ) and (REF ) hold with \(\Gamma ^{\prime }\)  in place of \(\Gamma .\)

By [9]}, \(\color {black}\Gamma ^r\)  (and hence \(\color {black}[\Gamma ^{\prime }]^r\) ) has a approximate tangent line at \(\mathcal {H}^1\) -a.e. \(x,\)  therefore, (REF ) follows from [7]}. To prove (REF ) with \(\Gamma \) 
choose \(x\in \Gamma \)  such that \(\theta (\Gamma ,x)=1\)  and \(T_x\)  exists. Without loss of generality we assume that \(x=0\)  and \(\nu _\Gamma (x)={\bf e_2}\)  is the unit normal to \(T_x.\) 
First we prove
\(\sigma _{0,r}(\Gamma ) \overset{K}{\rightarrow } T_0\) 
as \(r\searrow 0.\)  Indeed, let \(\mu _r:=\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\sigma _{0,r}(\Gamma ))\)  and \(\mu _0:=\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits T_0.\)  Given \(r>0,\)  since \(\mu _r( Q_\rho (x))=\frac{\mathcal {H}^1( Q_{\rho r}(rx))}{r},\)  by (REF ) for all \(x\in \sigma _{0,r}(\Gamma )\)  and \(\rho \in (0,r_0/r)\)  one has
\(c\le \frac{\mu _r( Q_\rho (x))}{2\rho } \le C.\) 
Let \(r_k\searrow 0\)  be any sequence. By compactness of sets in the Kuratowski convergence, passing to a further not relabelled subsequence if necessary we suppose that
\( \sigma _{0,r_k}(\Gamma ) \overset{K}{\rightarrow } L\) 
for some closed set \(L\subset \mathbb {R}^2\)  as \(k\rightarrow \infty .\)  We claim that \(L=T_0.\)  If there exists \(x\in T_0\setminus L,\)  then for some \(\rho >0,\)  \( Q_\rho (x)\cap L=\emptyset .\) 
By (REF ), \( Q_{\rho /2}(x)\cap \sigma _{0,r_k}(\Gamma )=\emptyset \)  for all large \(k\)  so that \(\mu _{r_k}( Q_{\rho /2}(x))=0.\)  Then by (REF )
\(0=\lim \limits _{k\rightarrow \infty } \mu _{r_k}( Q_{\rho /2}(x)) \ge \mu _0( Q_{\rho /2}(x))\ge \rho ,\) 
a contradiction. If there exists \(x\in L\setminus T_0,\)  then for some \( Q_\rho (x)\cap T_0=\emptyset \)  for some \(\rho >0\)  and there exists a sequence \(x_k\in \sigma _{0,r_k}(\Gamma )\)  such that \(x_k\rightarrow x.\)  Then \( Q_{\rho /2}(x_k)\subset Q_\rho (x)\)  for all large \(k\)  so that by (REF ) and (REF ),
\(0=\mu _0(\overline{ Q_\rho (x)}) \ge \limsup \limits _{k\rightarrow \infty } \mu _{r_k}(\overline{ Q_\rho (x)}) \ge \limsup \limits _{k\rightarrow \infty } \mu _{r_k}( Q_{\rho /2}(x_k))\ge c\rho ,\) 
a contradiction. Thus, \(L=T_0.\)  Since the sequence \(r_k\searrow 0\)  is arbitrary, (REF ) follows. Now (REF ) is obvious.
Next we turn to the compactness of displacements of the sequence of constrained minimizers \(\lbrace (A_m,u_m)\rbrace .\)
Proposition 4.4 
Let \(A_{m_h}\)  and \(A\)  be as in Proposition REF . Let \(\lbrace E_i\rbrace _{i\in \mathbb {N}}\)  be all connected components of \(\mathrm {Int}(A).\)  There exists a further (not relabelled) subsequence of \(\lbrace (A_{m_h},u_h)\rbrace \) , a sequence \(\lbrace a_h\rbrace \)  of rigid displacements, a subset \(N\)  of \(\mathbb {N}\) , a function \(v_0\in H^1(S)\)  and a family \(\lbrace v_i\in GSBD^2(\mathrm {Int}(E_i))\cap H_\mathrm {loc}^1(\mathrm {Int}(E_i)\cup S)\rbrace _{i\in N}\)  such that
\(|u_{m_h} + a_h | \rightarrow +\infty \) 
a.e. in \(\bigcup _{i\in \mathbb {N}\setminus N}E_i,\)
\(u_{m_h} + a_h \rightharpoonup v_0\chi _{S} + \sum \limits _{i\in N} v_i\chi _{E_i}\) 
weakly in \(H_\mathrm {loc}^1((\cup _{i\in N}E_i)\cup S)\)  (and hence a.e. in \((\cup _{i\in N}E_i)\cup S\) ),
\(e(u_{m_h}) \rightarrow e( v_0)\chi _{S} + \sum \limits _{i\in N} e(v_i)\chi _{E_i}\) 
weakly in \(L_\mathrm {loc}^2((\cup _{i\in N}E_i)\cup S).\)

The main difference of our compactness result from [2]} is not only that in our setting we have the set-function coupling, but also we need to select those components of limiting free crytal region where the displacements diverge and those in which they don't. This first requires to actually prove that the behavior is consistent inside each component of the limiting free-crystal region, which is achieved using [5]}.
Since \(S\)  is connected and Lipschitz, by the Korn-Poincaré inequality and the Rellich-Kondrachov Theorem there exists a further not relabelled subsequence \(\lbrace u_{m_h}\rbrace ,\)  a sequence \(\lbrace a_h\rbrace \)  of infinitesimal rigid displacements and \(v_0\in H^1(S;\mathbb {R}^2)\)  such that \(u_{m_h} + a_h \rightarrow v_0\)  weakly in \(H^1(S;\mathbb {R}^2)\)  and a.e. in \(S.\)
By a diagonal argument we choose a further not relabelled subsequence \(\lbrace u_{m_h}\rbrace \)  and the subset \(N\)  of indices \(i\in \mathbb {N}\)  for which there exists a ball \(B_i\subset \subset E_i\)  such that the limit of \(u_{m_h}(x) + a_h(x)\)  is finite for a.e. \(x\in B_i.\) 
We claim that for any \(i\in N\)  there exists \(v_i\in H^1_\mathrm {loc}(E_i;\mathbb {R}^2)\cap GSBD^2(E_i)\)  such that \(w_h:=u_{m_h}+ a_h \rightarrow v_i\)  weakly in \(H^1_\mathrm {loc}(E_i;\mathbb {R}^2)\)  and a.e. in \(E_i\)  as \(h\rightarrow \infty .\)
To prove the claim we fix \(i\in N\)  and let \(D\subset \subset E_i\)  be an arbitrary connected open set containing \(B_i.\)  Since \(\mathrm {sdist}(\cdot ,\partial A_{m_h})\rightarrow \mathrm {sdist}(\cdot ,\partial A)\)  locally uniformly in \(\mathbb {R}^2,\)  there exists \(h_D>0\)  such that \(D\subset \subset \mathrm {Int}(A_{m_h})\)  for all \(h>h_D.\) 
Note that \(w_h\in H^1(D)\)  and
\(\sup \limits _{h>h_D} \int _D |e(w_h)|^2dx \le C:=\frac{1}{2c_3}\sup \limits _{h>h_D} \int _{A_{m_h}\cup S} x)e(u_{m_h}):e(u_{m_h})dx<+\infty ,\) 
where in the first inequality we used (REF ) and in the second (REF ).
Since \(w_h\)  has finite limit a.e. in \(B_i\subset D,\)  by [5]} there exists \(v_i^D\in H_\mathrm {loc}^1(D)\cap GSBD^2(D)\)  and a subsequence \(\lbrace w_h^D\rbrace \)  of \(\lbrace w_h\rbrace \)  such that \(w_h^D \rightarrow v_i^D\)  weakly in \(H_\mathrm {loc}^1\)  and a.e. in \(D.\)  Now choosing a sequence \(D_1\subset \subset D_2\subset \subset \ldots \subset \subset E_i\)  of connected open sets such that \(B_i\subset D_1\)  and \(E_i = \cup _j D_j\)  and using a diagonal argument we choose a (not relablled) subsequence \(\lbrace w_h\rbrace \)  and \(v_i\in H_\mathrm {loc}^1(E_i)\cap GSBD_\mathrm {loc}^2(E_i)\)  such that \(w_h\rightarrow v_i\)  weakly in \(H_\mathrm {loc}^1(E_i)\)  and a.e. in \(E_i.\)  In particular, \(e(w_h)\rightarrow e(v_i)\)  weakly in \(L_\mathrm {loc}^2(E_i)\)  and hence, by convexity and (REF )
\(\int _{D_j} |e(v_i)|^2dx \le \liminf \limits _{h\rightarrow +\infty } \int _{D_j} |e(w_h)|^2dx \le C.\) 
Hence, letting \(j\rightarrow \infty \)  we get \(v_i\in GSBD^2(E_i).\)
By the choice of indices in \(N,\)  for any \(j\in \mathbb {N}\setminus N\)  one has \(|u_{m_h}+ a_h|\rightarrow +\infty \)  a.e. in \(E_j\)  as \(h\rightarrow +\infty .\)
Finally, since \(u_{m_h} + a_h \rightarrow u\)  weakly in \(H^1_\mathrm {loc}((\cup _{i\in N} E_i)\cup S),\)  it follows that \(e(u_{m_h})=e(u_{m_h} + a_h) \rightarrow e(u)\)  weakly in \(L^2_\mathrm {loc}((\cup _{i\in N} E_i)\cup S).\)
Proposition REF  allows us to define a “limit” displacement.
Proposition 4.5 
Let \(\lbrace (A_{m_h},u_{m_h})\rbrace ,\)  \(\lbrace a_h\rbrace ,\)  \(A,\)  \(N\)  and \(\lbrace v_i\rbrace _{i\in N\cup \lbrace 0\rbrace }\)  satisfy the assertion of Proposition REF  and let
\(u:=v_0 \chi _{S} + \sum \limits _{i\in N} v_i\chi _{E_i} + \sum \limits _{j\in \mathbb {N}\setminus N} u_0\chi _{E_j},\) 
where \(u_0\)  is the displacement defining the mismatch strain \(M_0\) . Then
\(\liminf \limits _{h\rightarrow \infty } \mathcal {W}(A_{m_h},u_{m_h})\ge \mathcal {W}(A,u).\) 


Fix arbitrary open set \(D\subset \subset \mathrm {Int}(A)\cup S.\)  By Proposition REF  \(u_{m_h} +a_h \rightarrow u\)  weakly in \(L^2(D\cap [(\cup _{i\in N} E_i) \cup S]),\)  hence, by the convexity of the elastic energy
\(&\liminf \limits _{h\rightarrow \infty } \mathcal {W}(A_{m_h},u_{m_h}) \color {black}=\liminf \limits _{h\rightarrow \infty } \int _{A_{m_h}\cup S} W(x,e(u_{m_h})-M_0)dx\nonumber \\\ge & \liminf \limits _{h\rightarrow \infty } \Big (\int _{D\cap S} W(x,e(u_{m_h}) - M_0)dx +\sum \limits _{j\in N} \int _{D\cap E_i} W(x,e(u_{m_h})-M_0)dx\Big )\nonumber \\\ge & \int _{D\cap S} W(x,e(u)-M_0)dx +\sum \limits _{i\in N} \int _{D\cap E_i} W(x,e(u) - M_0)dx,\) 

where we recall that \(M_0=e(u_0).\)  Since \(e(u) - M_0 =0\)  a.e. in \(\cup _{j\in \mathbb {N}\setminus N} E_j,\)  this inequality can also be rewritten as
\(\liminf \limits _{h\rightarrow \infty } \mathcal {W}(A_{m_h},u_{m_h})\ge \int _{D\cap (A\cup S)} W(x,e(u)-M_0)dx.\) 
Now letting \(D\nearrow \mathrm {Int}(A)\cup S\)  and using \(|A\setminus \mathrm {Int}(A)|\le |\partial A| =0\)  we get (REF ).
Now we establish the following “lower semicontinuity” of \(\mathcal {F}(A_m,u_m).\)
Proposition 4.6 
Let \(\lbrace (A_{m_h},u_{m_h})\rbrace ,\)  \(A\)  and \(u\)  be as in Proposition REF .
Then \((\mathrm {Int}(A),u)\in \widetilde{\mathcal {C}}\)  and
\(\liminf \limits _{h\rightarrow \infty } \mathcal {S}(A_{m_h},u_{m_h}) \ge \widetilde{\mathcal {S}}(\mathrm {Int}(A),u),\) 
where \(\widetilde{\mathcal {S}}\)  is defined in (REF ).

We postpone the proof of this proposition after the following auxiliary lemma, needed to treat the delamination and jumps along the cracks.
Lemma 4.7 
Let \(\phi \)  be any norm in \(\mathbb {R}^2.\)  Let \(\lbrace D_k\rbrace \)  and \(\lbrace m_k\rbrace \)  be sequences of subsets of \(Q_4\)  and of natural numbers, respectively, satisfying
(a)
the number of connected components \(\partial D_k\)  lying strictly inside \(Q_4\)  does not exceed \(m_k;\)
(b)
\(\mathrm {sdist}(\cdot ,\partial D_k)\rightarrow - \mathrm {dist}(\cdot ,I_4)\)  uniformly in \(Q_4\)  and
\(\sup \limits _k \mathcal {H}^1(Q_1\cap \partial D_k)<+\infty ;\) 
(c)
there exists a sequence \(\lbrace w_k\rbrace \subset GSBD^2(Q_4)\)  such that \(J_{w_k}\subset Q_1\cap \partial D_k\)  and
\(\sup \limits _k \int _{Q_1} |e(w_k)|^2dx<+\infty ;\) 
(d)
there exist \(\xi ^\pm \in \mathbb {R}^2\)  such that
\(w_k \rightarrow w_0:= \xi ^+ \chi _{Q_1^-} + \xi ^-\chi _{Q_1^+\setminus U_1^\infty }\quad \text{a.e. in $Q_1\setminus U_1^\infty $}\) 
and
\(|w_k|\rightarrow +\infty \quad \text{a.e. in $U_1^\infty ,$}\) 
where \(U_1^\infty \)  is either \(\emptyset \)  or \(Q_1^+.\)

Then there exists a subsequence \(\lbrace k_h\rbrace \subset \mathbb {N}\)  such that for any \(\delta \in (0,1)\)  we can find \(h_\delta >0\)  for which
\(\int _{Q_1\cap \partial ^*D_{k_h}} \phi (\nu _{D_{k_h}}) d\mathcal {H}^1 + 2\int _{Q_1\cap D_{k_h}^{(1)} \cap \partial D_{k_h}} \phi (\nu _{D_{k_h}})d\mathcal {H}^1 \ge 2\int _{I_1}\phi ({\bf e_2})d\mathcal {H}^1 -\delta \) 
for all \(h>h_\delta .\)

The proof is similar to [5]}.
Since \(\phi \)  is even,
\(\phi (\xi ) = \sup \limits _{\eta \in \mathbb {R}^2,\,\phi ^o(\eta )=1} |\xi \cdot \eta |,\quad \xi \in \mathbb {R}^2,\) 
where \(\phi ^o\)  is the dual norm of \(\phi .\)  By the compactness of \(B^{\phi ^o}:=\lbrace \eta \in \mathbb {R}^2:\,\phi ^o(\eta )=1\rbrace ,\)  for any countable set \(\lbrace \eta _i\rbrace \)  dense in \(B^{\phi ^o}\)  and for any \(\mathcal {H}^1\) -rectifiable set \(K\subset \mathbb {R}^2\)
\(\phi (\nu _K(x)) = \sup \limits _{i\ge 1} |\nu _K(x)\cdot \eta _i|\quad \text{for $\mathcal {H}^1$-a.e. $x\in K.$}\) 
Hence, by [15]} for any open set \(U\subset \mathbb {R}^2\)
\(\int _{U\cap K} \phi (\nu _K)d\mathcal {H}^1 = \sup \limits _k \sup \Big \lbrace \sum \limits _{i=1}^k\int _{A_i\cap K} |\nu _K\cdot \eta _i|d\mathcal {H}^1:\,\text{$A_i\subset \subset U$ open and pairwise disjoint}\Big \rbrace .\) 
Moreover, by the area formula for any Borel set \(B\)
\(\int _{B\cap K} |\nu _K\cdot \xi |d\mathcal {H}^1 = |\xi | \int _{\pi _\xi (B)} \mathcal {H}^0(K\cap B_y^\xi )d\mathcal {H}^1(y),\) 
where
\(\pi _\xi (z) = z - \big (z\cdot \tfrac{\xi }{|\xi |}\big )\,\tfrac{\xi }{|\xi |}\) 
and given \(y\in \pi _\xi (B),\)  \(B_y^\xi :=\pi _\xi ^{-1}(y)\cap B.\)
Before the proof of the lemma let us recall some notation and results from [2]}. Given \(\xi \in \mathbb {R}^2\setminus \lbrace 0\rbrace ,\)  let \(\Pi _\xi :=\lbrace y\in \mathbb {R}^2:\,y\cdot \xi =0\rbrace .\)  For every set \(B\subset \mathbb {R}^2\)  and for every \(y\in \Pi _\xi \)  we define
\(B_y^\xi :=\lbrace t\in \mathbb {R}:\, y+t\xi \in B\rbrace .\) 
Moreover, for every \(u:B\rightarrow \mathbb {R}^2\)  we define \(\widehat{u}_y^\xi :B_y^\xi \rightarrow \mathbb {R}\)  by
\(\widehat{u}_y^\xi (t):= u(y + t\xi )\cdot \xi .\) 
When \(u\in GSBD^2(Q_1),\)  then \(\widehat{u}_y^\xi \in SBV_\mathrm {loc}^2([Q_1]_y^\xi )\)  for \(\mathcal {H}^1\) -a.e. \(\pi _\xi (Q_1)\)  and for all \(\xi \in \mathbb {R}^2\setminus \lbrace 0\rbrace .\)  In this case we define
\(I_y^\xi (u):=\int _{[Q_1]_y^\xi } |(\dot{u})_y^\xi |^2 dt,\) 
where \((\dot{u})_y^\xi \)  is the density of the absolutely continuity part of \(D \widehat{u}_y^\xi \)  and also
\(II_y^\xi (u):=|D(\tau (u\cdot \xi )_y^\xi )|([Q]_1^\xi ),\) 
where \(\tau (t):=\arctan (t).\)  Recall that
\(\int _{\Pi _\xi } I_y^\xi (u) \mathcal {H}^1(y) +\int _{\Pi _\xi } II_y^\xi (u) \mathcal {H}^1(y)\le \int _{Q_1} |e(u)|dx + \int _{Q_1} |e(u)|^2dx + \mathcal {H}^1(J_u)\) 
(see e.g. [2]}).
Step 1: There exists an at most countable set \(\subset B^{\phi ^o}\)  such that
\(\lim \limits _{k\rightarrow +\infty } \mathcal {H}^1(\pi _\xi (I_1)\setminus \pi _\xi (J_{w_k})) =0\) 
for any \(\xi \in B^{\phi ^o}\setminus .\)
Indeed, let \(\)  be the set of all \(\xi \in B^{\phi ^o}\)  for which there exists \(y\in \pi _\xi (I_1)\)  such that \(\mathcal {H}^1(\pi _\xi ^{-1}(y)\cap \partial D_k)>0.\) 
By assumption (b) and Lemma REF  the set \(\)  is at most countable.
Let \(\lbrace w_{k_l}\rbrace \)  be arbitrary not relabelled subsequence of \(\lbrace w_k\rbrace .\)  In view of [2]} (applied with \(A=U_1^\infty \) ) for any \(\xi \in B^{\phi ^o}\setminus ,\)  \(\epsilon >0\)  and for \(\mathcal {H}^1\) -a.e. \(y\in \pi _\xi (Q_1)\)  there exists a further subsequence \(w_{k_{l_h}}\)  (possibly depending on \(\xi ,\)  \(\epsilon \)  and \(y\) )
\(\mathcal {H}^0(J_{[\widehat{w}_0]_y^\xi } \cap [Q_1\setminus U_1^\infty ]_y^\xi ) + \mathcal {H}^0([\partial U_1^\infty ]_y^\xi ) \le \liminf \limits _{h\rightarrow +\infty } \Big [\mathcal {H}^1(J_{[w_{k_{l_h}}]_y^\xi }) + \epsilon (I_y^\xi (w_{k_{l_h}}) + II_y^\xi {w_{k_{l_h}}})\Big ].\) 
By the definition of \(w_0\)  and \(U_1^\infty ,\)  the left-side of (REF ) is equal to 1 for \(\mathcal {H}^1\) -a.e. \(y\in \pi _\xi (I_1).\)  Theorefore, for such \(y\)  and for sufficiently small \(\epsilon >0\)  we have
\(\liminf \limits _{h\rightarrow +\infty } \mathcal {H}^1(J_{[w_{k_{l_h}}]_y^\xi })\ge 1.\)  Hence, for \(\mathcal {H}^1\) -a.e. \(y\in \pi _\xi (I_1)\)  the line \(\pi _\xi ^{-1}(y)\)  intersects \(J_{w_{k_{l_h}}}\)  for all \(h\)  and (REF ) follows.
Note that by [5]}
\(\liminf \limits _{k\rightarrow +\infty } \int _{Q_1\cap J_{w_k}} \phi (\nu _{J_{w_k}}) d\mathcal {H}^1 \ge \int _{I_1} \phi ({\bf e_2}) d\mathcal {H}^1.\) 
Step 2: Now we improve (REF ) by including coefficient 2 on the right-hand side of the inequality in the presence of a small error term.
We proceed in three substeps. We redefine the displacement \(w_k\)  in the convex envelope \(V_k^i\)  of each connected component \(K_k^i\)  of \(\partial D_k\)  in such a way that \(\partial V_k^i\)  become jump sets with the left-hand side of (REF ) lowered up to a small error.
Substep 2.1: First we identify \(\lbrace V_k^i\rbrace .\)
Fix any \(\delta \in (0,1).\) 
By (b) there exists \(k_\delta ^1>0\)  such that \(([-2,2]\times [-2,-\delta ]) \cup ([-2,2]\times [\delta ,2])\subset \mathrm {Int}(D_k)\)  for any \(k\ge k_\delta ^1.\)  Let \(F_k:=Q_1\cap D_k.\)  Note that \(\partial F_k \subset (Q_1\cap \partial D_k) \cup (\lbrace \pm 1\rbrace \times [-\delta ,\delta ])\)  and since \(D_k\in \mathcal {A}_{m_k},\)  the number of connected components \(\lbrace L_k^j\rbrace _{j\ge 1}\)  of \(\partial F_k\)  does not exceed \(m_k\) . Note that \(F_k\subset [-1,1]\times [-\delta ,\delta ]\)  and
\(\alpha _k: = & \int _{Q_1\cap \partial ^*E_{k}} \phi (\nu _{E_{k}}) d\mathcal {H}^1 + 2\int _{Q_1\cap E_{k}^{(1)} \cap \partial E_{k}} \phi (\nu _{E_{k}})d\mathcal {H}^1 \nonumber \\\ge & \int _{Q_2\cap \partial ^*F_k} \phi (\nu _{F_k}) d\mathcal {H}^1 + 2\int _{Q_2\cap F_k^{(1)} \cap \partial F_k} \phi (\nu _{F_k})d\mathcal {H}^1 - 4\delta \nonumber \\= & \sum \limits _{j\ge 1} \Big [\int _{Q_2\cap \partial ^*F_k\cap L_k^j} \phi (\nu _{F_k}) d\mathcal {H}^1 + 2\int _{Q_2\cap F_k^{(1)} \cap \partial F_k\cap L_k^j} \phi (\nu _{F_k})d\mathcal {H}^1\Big ] - 4\delta :=\alpha _k^{\prime }.\) 

Next repeating the same arguments of Step 1 in the proof of [5]} we can find a family \(\lbrace V_k^i\rbrace _i\)  of at most countably many pairwise disjoint closed convex sets with non-empty interior such that for each \(L_k^j\)  there exists a unique \(V_i\)  with \(L_k^j\subset V_k^i\)  and
\(\alpha _k^{\prime } \ge \sum \limits _{i\ge 1} \int _{\partial V_k^i} \phi (\nu _{V_k^i}^{}) d\mathcal {H}^1 - 6\delta \) 
see e.g. [5]}
Substep 2.2: Now we replace \(w_k\)  with another function \(v_k\)  associated to \(V_k^i.\)  Fix \(\xi _0\in \mathbb {R}^2\)  such that the jump set of the function
\(v_k:= w_k\chi _{Q_1\setminus \cup _i V_k^i} + \xi _0\chi _{\cup _i V_k^i}\) 
coincide with \(\cup _i \partial V_k^i\)  (up to a \(\mathcal {H}^1\) -negligible set).
By assumption (b) \(\cup _i\partial V_k^i \overset{K}{\rightarrow }I_1\)  as \(k\rightarrow +\infty .\)  Moreover, as in Step 1 we can find a countable set \(^{\prime }\subset B^{\phi ^o}\)  such that
by assumption (b) and (REF )
\(\limsup \limits _{k\rightarrow +\infty } \mathcal {H}^1(\pi _\xi (I_1)\setminus \pi _\xi (\cup _i\partial V_k^i)) \le & \limsup \limits _{k\rightarrow +\infty } \mathcal {H}^1(\pi _\xi (I_1)\setminus \pi _\xi (\partial D_k)) \nonumber \\\le &\lim \limits _{k\rightarrow +\infty } \mathcal {H}^1(\pi _\xi (I_1)\setminus \pi _\xi (J_{w_k})) =0\) 

for all \(\xi \in B^{\phi ^o}\setminus (\cup ^{\prime }).\) 
Moreover, by assumption (d) \(v_k \rightarrow w_0\)  a.e. in \(Q_1\setminus U_1^\infty \)  and \(|v_k|\rightarrow +\infty \)  a.e. in \(U_1^\infty .\)
Substep 2.3: By convexity of each \(V_k^i\)  we observe that
\(\liminf \limits _{k\rightarrow +\infty } \mathcal {H}^0(\pi _\xi ^{-1}(y) \cap J_{v_k}) \ge 2 = 2\mathcal {H}^0(J_{[\widehat{w}_0]_y^\xi } \cap [Q_1\setminus U_1^\infty ]_y^\xi ).\) 
for all \(\xi \in B^{\phi ^o}\setminus (\cup ^{\prime })\)  and \(\mathcal {H}^1\) -a.e. \(y\in \pi _\xi .\) 
Thus, by repeating the arguments of Step 1 in the proof of [5]} we get
\(\liminf \limits _{k\rightarrow +\infty } \int _{\cup _i \partial V_k^i} \phi (\nu _{\cup _iV_k^i}^{})d \mathcal {H}^1 = \liminf \limits _{k\rightarrow +\infty } \int _{J_{v_k}} \phi (\nu _{J_{v_k}}^{})d \mathcal {H}^1 \ge 2\int _{I_1} \phi ({\bf e_2})d\mathcal {H}^1,\) 
which together with (REF ) and (REF ) implies the assertion of the lemma.
Now we are ready to prove (REF ).
For shortness, let
\(G:=\mathrm {Int}(A).\) 
We define
\(\widetilde{u}_h := (u_{m_h} + a_h)\chi _{A_{m_h}}^{} + \eta \chi _{\Omega \setminus A_{m_h}}\) 
and
\(\widetilde{u}:= u\chi _G^{} + \eta \chi _{\Omega \setminus G}^{}\) 
for \(\eta \in (0,1)^2\)  such that \(\Omega \cap \partial ^*A_{m_h}\subset J_{\widetilde{u}_h}\)  and \(\Omega \cap \partial ^*G\subset J_{\widetilde{u}}\)  up to an \(\mathcal {H}^1\) -negligible set. Such \(\eta \)  exists by Lemma REF  in view of the estimate
\(\mathcal {H}^1(\partial A_{m_h}) \le \frac{\mathcal {S}(A_{m_h},u_{m_h})}{c_1} + \frac{2c_2\mathcal {H}^1(\partial \Omega )}{c_1}\le \frac{\mathcal {F}(A_1,u_1)}{c_1} + \frac{2c_2\mathcal {H}^1(\partial \Omega )}{c_1},\) 
which holds for every \(h\ge 1.\) 
Notice that \(\widetilde{u}_h\in GSBD^2({\rm Int}{(\Omega \cup S\cup \Sigma )})\cap H_\mathrm {loc}^1((\Omega \cup S) \setminus \partial A_{m_h}),\)  \(\widetilde{u}\in GSBD^2({\rm Int}{(\Omega \cup S\cup \Sigma )})\cap H_\mathrm {loc}^1((\Omega \cup S) \setminus \partial G),\)  \(J_{\widetilde{u}}\subset (\Omega \cap \partial G)\cup (\Sigma \cap J_u)\)  and
\(\mathcal {H}^1(J_{\widetilde{u}_h}) + \int _\Omega |e(\widetilde{u}_h)|^2dx\le \mathcal {F}(A_{m_h},u_{m_h}) + \mathcal {H}^1(\Sigma )\le M:= \mathcal {F}(A_1,u_1) + \mathcal {H}^1(\Sigma )<\infty \) 
for every \(h\ge 1.\)  Moreover, by Proposition REF , the definitions of \(u,\)  \(\widetilde{u}_h\)  and \(\widetilde{u},\)
\(\widetilde{u}_h \rightarrow \widetilde{u} \quad \text{a.e. in $[S\cup (\Omega \setminus G)]\cup \bigcup \limits _{i\in N} E_i$}\) 
and
\(|\widetilde{u}_h| \rightarrow +\infty \quad \text{a.e. in $\bigcup \limits _{j\in \mathbb {N}\setminus N} E_j,$}\) 
where \(\lbrace E_i\rbrace \)  and \(N\)  are provided by Proposition REF .
We observe that a priori \(\partial A,\)  and hence \(\partial G\) , does not need to be \(\mathcal {H}^1\) -rectifiable. Therefore, by [23]} \(J_{\widetilde{u}}\subset (\Omega \cap \partial ^r G)\cup (\Sigma \cap J_u),\)  where we recall that \(\partial ^rG\)  is \(\mathcal {H}^1\) -rectifiable part of \(\partial G.\)
To prove (REF ) we use similar arguments as in [5]}. Let
\(g \in L^\infty (\Sigma \times \lbrace 0,1\rbrace )\)  be such that
\(g(x,s): = \varphi (x,\nu _\Sigma (x)) + s\beta (x)\) 
for which we know by (REF ) that \(g\ge 0\)  and
\(|g(x,1) - g(x,0)| \le \varphi (x,\nu _\Sigma (x))\qquad \text{for a.e. $x\in \Sigma .$ }\) 
Let \(\mu _h\)  be the sequence of positive Radon measures defined at Borel sets \(B\subset \mathbb {R}^2\)  as
\(\begin{aligned}\mu _h(B) &: = \int _{B\cap \Omega \cap \partial ^*A_{m_h}} \varphi (x,\nu _{A_{m_h}})d\mathcal {H}^1 + 2\int _{B\cap \Omega \cap (A_{m_h}^{(1)}\cup A_{m_h}^{(0)})\cap \partial A_{m_h}}\varphi (x,\nu _{A_{m_h}})d\mathcal {H}^1\nonumber \\& + \int _{B\cap \Sigma \cap A_{m_h}^{(0)}\cap \partial A_{m_h} } \big [\varphi (x,\nu _\Sigma )+ g(x,1)\big ]d\mathcal {H}^1(x) + \int _{B\cap \Sigma \setminus \partial A_{m_h}} g(x,0)d\mathcal {H}^1 \nonumber \\& + \int _{B\cap \Sigma \cap \partial ^*A_{m_h} \setminus J_{u_{m_h}} } g(x,1) d\mathcal {H}^1+ \int _{B\cap \Sigma \cap J_{u_{m_h}}} \big [g(x,0) + \varphi (x,\nu _\Sigma )\big ]\,d\mathcal {H}^1\end{aligned}\) 
and
let \(\mu \)  be the positive measure defined at Borel sets \(B\subset \mathbb {R}^2\)  as
\(\begin{aligned}& \mu (B) : = \int _{B\cap \Omega \cap \partial ^*G} \varphi (x,\nu _A)d\mathcal {H}^1 +2\int _{B\cap \Omega \cap G^{(1)} \cap \partial G\cap J_{\widetilde{u}}} \varphi (x,\nu _G)d\mathcal {H}^1\\+& \int _{B\cap \Sigma \setminus \partial G} g(x,0)d\mathcal {H}^1 + \int _{B\cap \Sigma \cap \partial ^*G \setminus J_u} g(x,1) d\mathcal {H}^1+ \int _{B\cap \Sigma \cap J_{\widetilde{u}}} \big [g(x,0) + \varphi (x,\nu _\Sigma )\big ]\,d\mathcal {H}^1.\end{aligned}\) 
Since \(S_{\widetilde{u}}^A:=G^{(1)}\cap \partial G\cap J_{\widetilde{u}}\)  and \(\Sigma \cap J_{\widetilde{u}} = \Sigma \cap J_u,\) 
we have
\(\mu _h(\mathbb {R}^2) = \mathcal {S}(A_{m_h} ,u_{m_h}) + \int _{\Sigma } \varphi (x,\nu _\Sigma )d\mathcal {H}^1\) 
and
\(\mu (\mathbb {R}^2) = \widetilde{\mathcal {S}}(G,u) + \int _{\Sigma } \varphi (x,\nu _\Sigma )d\mathcal {H}^1.\) 
Hence, to establish (REF ) it suffices to prove
\(\liminf \limits _{h\rightarrow \infty } \mu _h(\mathbb {R}^2) \ge \mu (\mathbb {R}^2).\) 
Since \(\sup _h \mu _h(\mathbb {R}^2)<+\infty ,\)  by compactness, there exists a positive Radon measure \(\mu _0\)  in \(\mathbb {R}^2\)  such that (up to a subsequence) \(\mu _h\rightharpoonup ^* \mu _0\)  as \(h\rightarrow \infty .\) 
We show
\(\mu _0\ge \mu \) 
and we observe that (REF ) immediately follows from (REF ).
To establish (REF ) it suffices to prove
\(& \frac{d\mu _0}{d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Omega \cap \partial ^*G)}(x) \ge \varphi (x,\nu _G(x))\qquad \text{for a.e. $x\in \Omega \cap \partial ^*G,$}\\& \frac{d\mu _0}{d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Sigma \cap \partial ^* G)}(x) \ge g(x,1) \qquad \text{for a.e. $x\in \Sigma \cap \partial ^* G,$}\\& \frac{d\mu _0}{d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Sigma \setminus \partial G)}(x) \ge \varphi (x,\nu _\Sigma (x)) \qquad \text{for a.e. $x\in \Sigma \setminus \partial G,$} \\& \frac{d\mu _0}{d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits S_{\widetilde{u}}^A }(x) \ge 2\varphi (x,\nu _G(x))\qquad \text{for a.e. $x\in S_{\widetilde{u}}^A,$ }\\& \frac{d\mu _0}{d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits (\Sigma \cap J_{\widetilde{u}})}(x) \ge 2\varphi (x,\nu _\Sigma (x)) \qquad \text{for a.e. $x\in \Sigma \cap J_{\widetilde{u}}$}\) 


since \(g(x,0)=\varphi (x,\nu _\Sigma ).\) 
We use similar arguments of the proof of [5]}.
The proof of the estimates (REF )-() follows from similar arguments used in [5]} with special care needed for (). In fact for () we cannot employ the strategy used for [5]} which was hinged on the uniform bound on the number of boundary components, which here we do not have. We instead adapt the arguments employed in [5]} by using Lemma REF .
Next we detail the proofs of (REF )-().
Proof of (REF ). Note that \(A =G\)  up to a negligible set.
By Corollary REF  \(A_{m_h}\rightarrow A\)  in \(L^1(\mathbb {R}^2),\)  thus, the proof of (REF ) can be done following the arguments of [5]} using Reshetnyak lower semicontinuity Theorem [7]}.
Proof of ().
Since \(A_{m_h} \rightarrow G\)  in \(L^1(\mathbb {R}^2),\)  we have \(D\chi _{A_{m_h}}\rightharpoonup ^* D\chi _G.\)  Thus, the proof of () directly follows from [31]} (see also the proof of [5]}).
Proof of ().
Let \(x_0\in \Sigma \setminus \partial G\)  and let \(r_0:=\mathrm {dist}(x_0,\partial G) >0.\)  Since \(\mathbb {R}^2\setminus \mathrm {Int}(A_{m_h})\overset{K}{\rightarrow } \mathbb {R}^2\setminus \mathrm {Int}(A)=\mathbb {R}^2\setminus G,\)  there exists \(r_1\in (0,r_0)\)  such that \(B_r(x_0)\cap \mathrm {Int}(A_{m_h})\)  for any \(r\in (0,r_1).\)  Therefore, for any \(r\in (0,r_1)\)
\(\begin{aligned}\mu _h(\overline{B_r(x_0)}) = & \int _{\overline{B_r(x_0)}\cap \Sigma \cap A_{m_h}^{(0)}\cap \partial A_{m_h} } \big [\varphi (x,\nu _\Sigma )+ g(x,1)\big ]d\mathcal {H}^1(x) + \int _{\overline{B_r(x_0)}\cap \Sigma \setminus \partial A_{m_h}} g(x,0)d\mathcal {H}^1\\\ge & \int _{\overline{B_r(x_0)}\cap \Sigma \cap A_{m_h}^{(0)}\cap \partial A_{m_h} } g(x,0)d\mathcal {H}^1(x) + \int _{\overline{B_r(x_0)}\cap \Sigma \setminus \partial A_{m_h}} g(x,0)d\mathcal {H}^1 \\= & \int _{\overline{B_r(x_0)}\cap \Sigma } g(x,0)d\mathcal {H}^1,\end{aligned}\) 
where in the inequality we used (REF ).
Thus, taking \(limsup\)  as \(h\rightarrow +\infty \)  and using \(\mu _h \rightharpoonup ^*\mu _0\)  we get
\(\mu _0(\overline{B_r(x_0)}) \ge \int _{B_r(x_0)\cap \Sigma } g(x,0)d\mathcal {H}^1.\) 
Now () follows from the Besicovitch Differentiation Theorem.
Proofs of () and (). We establish
\(\frac{d\mu _0}{d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits K } \ge 2\phi (x,\nu _K) \quad \text{for $\mathcal {H}^1$-a.e. $x\in K,$}\) 
where
\(K=S_{\widetilde{u}}^A \cup (\Sigma \cap J_{\widetilde{u}}).\) 
Let \(x\in K\)  be such that \(\theta (K,x)=1.\)  Then either \(x\in S_{\widetilde{u}} \subset G^{(1)}\cap \partial ^r G\)  or \(x\in \Sigma \cap J_{\widetilde{u}}.\)  By setting \(E_0:=S\)  and recallling that \(\mathrm {Int}(A)=\cup _{i\in \mathbb {N}} E_i,\)  in view of Proposition REF 
we have the one of the following:
(b1)
there exists \(i_0\in N\)  such that \(x\in E_{i_0}^{(1)}\cap \partial ^r E_{i_0},\) 
\(\theta (E_{i_0}^{(1)}\cap \partial ^r E_{i_0},x)=1\) 
and \(u_{m_h} + a_{m_h} \rightarrow u\)  a.e. in \(E_{i_0};\)
(b2)
there exist \(i_0\in N\cup \lbrace 0\rbrace \)  and \(j_0\in \mathbb {N}\setminus N\) 
such that \(x\in \partial ^*E_{i_0}\cap \partial ^* E_{j_0}\)  and \(u_{m_h} + a_{m_h}\rightarrow u\)  a.e. in \(E_{i_0}\)  and \(|u_{m_h} + a_{m_h}|\rightarrow \infty \)  a.e. in \(E_{j_0};\)
(b3)
there exist \(i_1,i_2\in N\cup \lbrace 0\rbrace \)  with \(i_1\ne i_2\)  such that \(x\in \partial ^*E_{i_1}\cap \partial ^* E_{i_2}\)  and \(u_{m_h} + a_{m_h}\rightarrow u\)  a.e. in \(E_{i_1}\cup E_{i_2}.\)

Let \(L\)  denote the set among \(E_{i_0}^{(1)}\cap \partial ^r E_{i_0},\)  \(\partial ^*E_{i_0}\cap \partial ^* E_{j_0}\)  and \(\partial ^*E_{i_1}\cap \partial ^* E_{i_2}\)  containing \(x.\)  Without loss of generality we assume that \(x\in Y\subset L,\) 
where \(Y\)  is defined as the set of points \(y\in L\subset \partial A\)  satisfying:
(c1)
\(\theta (\partial G,y) = \theta (\partial A,y) =\theta (L,y)= 1\)  and \(\nu _G(y)=\nu _A(y)=\nu _L(y)\)  exists. If \(y\in \Sigma ,\)  then additionally, \(\theta (\Sigma ,x) =1\)  and \(\nu _\Sigma \)  also exists;
(c2)
as \(\rho \rightarrow 0\)  the sets \(\overline{ Q_{R,\nu _L}(y)}\cap \sigma _{\rho ,x}(\partial A), \)  \(\overline{ Q_{R,\nu _L}(x)}\cap \sigma _{\rho ,y}(\partial G) \)  and \(\overline{ Q_{R,\nu _L}(y)}\cap \sigma _{\rho ,y}(L) \)  converge \(\overline{Q_{R,\nu _L}(y)} \cap T_y\)  in the Kuratowski sense, where \(R>0\)  and \(T_y\)  is the generalized tangent line to \(\partial A\)  at \(y;\)
(c3)
one-sided traces \(\widetilde{u}^+(y)\)  and \(\widetilde{u}^-(y)\)  of \(\widetilde{u}\)  w.r.t. \(L\)  exist and are not equal;
(c4)
\(\frac{d\mu _0}{ d\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits K}\,(y)\)  exists and is finite.

In fact, \(\mathcal {H}^1(L\setminus Y)=0\)  since for (c1) we notice that \(Y\subset L \subset \partial ^rA\)  and \(\partial ^rA\)  is \(\mathcal {H}^1\) -rectifiable, for (c2) we use Proposition REF  by observing that the points of \(\Sigma \)  and \(\Omega \cap \partial A\)  satisfy uniform density estimates in view of the Lipchitzianity of \(\Sigma \)  and Proposition REF , respectively, for (c3) we use
[23]} and the existence of traces of \(GBD\) -functions along \(C^1\) -manifolds [23]} and the fact that being a jump set of \(\widetilde{u},\)  the set \(K\)  (and also \(L\) ) can be covered by at most countably many one-dimensional \(C^1\) -graphs (up to a \(\mathcal {H}^1\) -negligible set), and finally for (c4) we use Besicovitch Differentiation Theorem.
Without loss of generality, we assume \(x=0,\)  \(\nu _K(x)={\bf e_2},\)  \(T_x\)  is the \(x_1\) -axis and \({\bf e_2}\)  is the outer normal of \(E_{i_0}\) .
Let \(4r_0:=\mathrm {dist}(0,\partial \Omega )\)  if \(0\in \Omega \)  and \(4r_0:=\mathrm {dist}(0,\partial \Sigma )\)  if \(0\in \Sigma ;\)  since \(\Sigma \)  is Lipschitz, it consists of at most countably many open connected components in \(\partial \Omega ,\)  and hence, \(r_0>0\) . By weak convergence,
\(\lim \limits _{h\rightarrow \infty } \mu _h(\overline{ Q_r}) = \mu _0( Q_r)\) 
for a.e. \(r\in (0,r_0).\)  By assumption (b3), [23]} and [23]} separately applied to \( Q_1^+:=Q_1\cap \lbrace x_2>0\rbrace \)  and \( Q_1\setminus Q_1^+\)  we have
\(\lim \limits _{r\rightarrow 0} \int _{ Q_1} |\tau (\widetilde{u}(rx)) - \tau (u_0(x))| dx =0,\) 
where
\(u_0 := \widetilde{u}^+(0)\chi _{ Q_1^+} + \widetilde{u}^-(0)\chi _{ Q_1\setminus Q_1^+}\) 
and
\(\tau (z) = (\arctan z_1,\arctan z_2),\quad z=(z_1,z_2)\in \mathbb {R}^2.\) 
For every \(r\in (0,r_0)\)  let
\(U_r^\infty :=\lbrace x\in Q_1:\,\, \liminf \limits _{h\rightarrow \infty } |\widetilde{u}_h(rx)|=+\infty \rbrace .\) 
Unlike the proof of [5]}, (REF ) implies that \(U_r^\infty \)  can have positive measure.
By (REF ) and the Dominated Convergence Theorem
\(\lim \limits _{h\rightarrow \infty } \int _{ Q_1\setminus U_r^\infty } |\tau (\widetilde{u}_h(rx)) - \tau (\widetilde{u}(rx))| dx = 0.\) 
By (c2) applied with \(R=8\) , Proposition REF  and (c1)-(c3)
\(& Q_8\cap \sigma _r(\partial A) \overset{K}{\rightarrow } I_8\qquad \text{and}\qquad \mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits ( Q_8\cap \sigma _r(\partial A)) \overset{*}{\rightharpoonup } \mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits I_8, \\& Q_8\cap \sigma _r(L) \overset{K}{\rightarrow } I_8\quad \text{and}\quad \mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits ( Q_8\cap \sigma _r(L) )) \overset{*}{\rightharpoonup } \mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits I_8 \) 

as \(r\rightarrow 0.\)  Hence, by [5]}
\(& \mathrm {sdist}(\cdot , \sigma _r(\partial A)) \rightarrow - \mathrm {dist}(\cdot ,T_0), \\& \mathrm {sdist}(\cdot , \sigma _r(\partial E_{i_0})) \rightarrow -\mathrm {dist}(\cdot ,T_0),\\&\mathrm {sdist}(\cdot , \sigma _r(\partial [E_{i_0}\cup E_{j_0}])) \rightarrow -\mathrm {dist}(\cdot ,T_0), \\&\mathrm {sdist}(\cdot , \sigma _r(\partial [E_{i_1}\cup E_{i_2}])) \rightarrow -\mathrm {dist}(\cdot ,T_0) \) 


locally uniformly in \(\overline{ Q_4}\)  as \(r\rightarrow 0.\) 
Let
\(U_0^\infty ={\left\lbrace \begin{array}{ll}\emptyset & \text{in cases (c1) and (c3)},\\Q_1^+ & \text{in case (c2)}.\end{array}\right.}\) 
By the definitions of \(E_{i_0},\)  \(E_{j_0},\)  \(E_{i_1}\)  and \(E_{i_2}\)  and ()-()
\(\lim \limits _{r\rightarrow 0} | U_r^\infty \Delta U_0^\infty | =0.\) 
Step 1: We choose sequences \(h_k\nearrow \infty \)  and \(r_k\searrow 0\)  as follows. By (REF ), (REF ), (REF ) and (REF ) for any \(k\in \mathbb {N}\)  there exists \(r_k\in (0,\frac{1}{k})\)  such that (REF ) holds with \(r=r_k\)  and
\(&\Vert \mathrm {sdist}(\cdot , \sigma _{r_k} (\partial A)) + \mathrm {dist}(\cdot , T_0) \Vert _{L^\infty ( Q_4)} <\frac{1}{k^2},\\&\int _{ Q_1} |\tau (\widetilde{u}(r_k x)) - \tau (u_0(x))| dx <\frac{1}{k^2}, \\&| U_{r_k}^\infty \Delta U_0^\infty | < \frac{1}{k^2} .\) 

Given \(k\ge 1\)  and \(r_k\) , since \(A_{m_h}\)  \(sdist\) -converges to \(A\)  and the function \(\tau \)  is bounded, by (REF ), () and (REF ) we can choose \(h_k\)  such that
\(&\frac{1}{h_kr_k} <\frac{1}{k},\\&\Vert \mathrm {sdist}(\cdot , \sigma _{r_k} (\partial A_{m_{h_k}} )) - \mathrm {sdist}(\cdot , \sigma _{r_k} ( \partial A))\Vert _{L^\infty ( Q_4)} <\frac{1}{k},\\&\int _{ Q_1\setminus U_0^\infty } |\tau (\widetilde{u}_{h_k}(r_kx)) - \tau (\widetilde{u}(r_kx))| dx < \frac{1}{k},\\& \mu _{h_k}(\overline{ Q_{r_k}}) \le \mu _0( Q_{r_k}) +r_k^2. \) 


Notice that by (REF ), \(h_k\rightarrow \infty \)  as \(k\rightarrow \infty .\)
Let
\(D_k:= \sigma _{r_k} (A_{m_{h_k}}\cup S)\) 
and
\(w_k(x):=\widetilde{u}_{h_k}(r_kx),\quad x\in Q_1.\) 
Then the number of connected components of \(\partial D_k\)  lying strictly inside \(Q_4\)  does not exceed \(m_{h_k},\)  and \(w_k\in GSBD^2(Q_1)\)  with \(J_{w_k}\subset Q_1\cap \partial D_k.\) 
By ()
and (REF ),
\(\mathrm {sdist}(\cdot ,\partial D_k) \rightarrow -\mathrm {dist}(\cdot , T_0) \quad \text{uniformly in $Q_4$ as $k\rightarrow \infty .$}\) 
Moreover, by () and () \(w_k\rightarrow u_0\)  a.e. in \(Q_1\setminus U_1^\infty \)  and \(|w_k|\rightarrow +\infty \)  a.e. in \(U_1^\infty .\) 
By the finiteness of
\(\frac{d\mu _0}{\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits L}(0) = \lim \limits _{k\rightarrow \infty } \frac{\mu _0( Q_{r_k})}{2r_k}\) 
and ()
\(\mathcal {H}^1( Q_1\cap \partial D_k) =\frac{\mathcal {H}^1( Q_{r_k}\cap \partial A_{m_{h_k}} )}{r_k} \le \frac{\mu _{h_k}(\overline{ Q_{r_k}})}{c_1 r_k} \le C:=\frac{2}{c_1}\frac{d\mu _0}{\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits L}(0) + 1\) 
for all large \(k.\) 
Moreover, by changing variables as \(x=r_ky\)  and using (REF ) we get
\(\int _{Q_1} |e(w_k)|^2dx = \int _{Q_{r_k}} |e(\widetilde{u}_k)|^2dy \le M\) 
for all \(k;\)  note that the first equality holds only in dimension two.
Fix \(\delta \in (0,1).\) 
Since \(\varphi \)  is uniformly continuous, there exists \(k_\delta ^0>0\)  such that
\(|\varphi (x,\nu ) - \varphi (0,\nu )| < \delta ,\qquad x\in \overline{ Q_{r_k}},\,\,\nu \in \mathbb {S}^1.\) 
Therefore, by the definitions of \(D_k\)  and \(\mu _h,\)  the nonnegativity of \(g\)  as well as (REF )
\(\frac{\mu _{h_k}(\overline{ Q_{r_k}})}{r_k} \ge &\int _{ Q_1\cap \partial ^* D_k} \phi (\nu _{D_k})d\mathcal {H}^1+ 2\int _{ Q_1\cap D_k^{(1)} \cap \partial D_k} \phi (\nu _{D_k})d\mathcal {H}^1 - 2Cc_2\delta ,\) 

where
\(\phi (\nu ) = \varphi (0,\nu ).\) 
By Lemma REF  applied with sequences \(\lbrace D_k\rbrace \)  and \(\lbrace m_{h_k}\rbrace \)  we find \(k_\delta ^2>k_\delta ^1\)  such that
\(\int _{ Q_1\cap \partial ^* D_k} \phi (\nu _{D_k})d\mathcal {H}^1+ 2\int _{ Q_1\cap D_k^{(1)} \cap \partial D_k} \phi (\nu _{D_k})d\mathcal {H}^1 \ge 2\int _{I_1} \phi ({\bf e_2})d\mathcal {H}^1 - \delta .\) 
Thus, by (REF ) and () we get
\(\frac{\mu _0(Q_{r_k})}{2r_k} + \frac{r_k}{2} \ge \int _{I_1} \phi ({\bf e_2})d\mathcal {H}^1 - \frac{2Cc_1 + 1}{2}\,\delta \) 
for all \(k>k_\delta ^2.\)  Now letting first \(k\rightarrow +\infty \)  and then \(\delta \rightarrow 0\)  we get (REF ).

Proof of the main results
The aim of this section is to prove theorems of Section REF .
We start by showing that the volume-constraint infima of \(\mathcal {F}\)  in \(\mathcal {C}\)  and of \(\widetilde{\mathcal {F}}\)  in \(\widetilde{\mathcal {C}}\)  in fact coincide.
Proposition 5.1 
Assume hypotheses (H1)-(H3) and let \(\mathtt {v}\in (0,|\Omega |)\)  or \(S=\emptyset .\) 
Then
\(\inf \limits _{(A,u)\in \mathcal {C},\,|A|=\mathtt {v}}\, \mathcal {F}(A,u)\,\, = \inf \limits _{(A,u)\in \widetilde{\mathcal {C}},\,|A|=\mathtt {v}}\, \widetilde{\mathcal {F}}(A,u) = \inf \limits _{(A,u)\in \widetilde{\mathcal {C}}}\, \widetilde{\mathcal {F}}^\lambda (A,u)\) 
for any \(\lambda \ge \lambda _0,\)  where \(\lambda _0\)  is given by [5]} and \(\widetilde{\mathcal {F}}^\lambda \)  is given by (REF ).

We repeat similar arguments to [5]}.
Note that for any \(\lambda >0\)
\(\inf \limits _{(A,u)\in \mathcal {C},\,|A|=\mathtt {v}}\, \mathcal {F}(A,u)\,\, \ge \inf \limits _{(A,u)\in \widetilde{\mathcal {C}},\,|A|=\mathtt {v}}\, \widetilde{\mathcal {F}}(A,u) \ge \inf \limits _{(A,u)\in \widetilde{\mathcal {C}}}\, \widetilde{\mathcal {F}}^{\lambda }(A,u).\) 
Further we fix any \(\lambda \ge \lambda _0.\) 
Recall that from [5]} for such \(\lambda \)
\(\inf \limits _{(A,u)\in \mathcal {C},\,|A|=\mathtt {v}}\, \mathcal {F}(A,u) =\lim \limits _{m\rightarrow +\infty }\,\min \limits _{(A,u)\in \mathcal {C}_m,\,|A|=\mathtt {v}}\, \mathcal {F}(A,u)=\lim \limits _{m\rightarrow +\infty }\,\min \limits _{(A,u)\in \mathcal {C}_m}\, \mathcal {F}^{\lambda }(A,u),\) 
where \(\mathcal {F}^\lambda \)  is given by (REF ). Thus, in view of (REF ) to prove (REF ) it is enough to establish that \(\epsilon >0\)  there exists \(n_\epsilon \in \mathbb {N}\)  and \((A_\epsilon ,u_\epsilon )\in \mathcal {C}_{n_\epsilon }\)  such that
\(\inf \limits _{(A,u)\in \widetilde{\mathcal {C}}}\, \widetilde{\mathcal {F}}^{\lambda }(A,u) + \epsilon > \mathcal {F}^{\lambda }(A_\epsilon ,u_\epsilon ).\) 
To prove the existence of \(n_\epsilon \)  and \((A_\epsilon ,u_\epsilon )\in \mathcal {C}_{n_\epsilon }\) , we repeat essentially the same arguments of the proof of [5]}. For the convenience of the reader we give the detailed proof. Given \(\epsilon >0\)  let \((B_1,v_1)\in \widetilde{\mathcal {C}}\)  be such that
\(\inf \limits _{(A,u)\in \widetilde{\mathcal {C}}}\, \mathcal {F}^{\lambda }(A,u)> \mathcal {F}^{\lambda }(B_1,v_1) -\epsilon .\) 
Since \(|B_1| = |\mathrm {Int}(B_1)|\)  and \(\mathcal {F}^{\lambda }(B_1,v_1)\ge \mathcal {F}^{\lambda }(\mathrm {Int}(B_1),v_1),\)  we may assume that \(B_1=\mathrm {Int}(B_1),\)  i.e., \(B_1\)  is open.
Step 1: First we remove the jump set \(J_{v_1}\)  of \(v_1\)  on \(\Sigma \)  making a hole in \(\Omega .\)  Recall that by our choice, \(\nu _\Sigma \)  is always directed towards \(\Omega .\)  Since \(\Sigma \)  is Lipschitz, by the regularity of \(\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits \Sigma ,\)  there exists an open set \(\Sigma ^{\prime }\subset \Sigma \)  such that \(\mathcal {H}^1(J_{v_1}\setminus \Sigma ^{\prime })=0\)  and \(\mathcal {H}^1(\Sigma ^{\prime }\setminus J_{v_1})<\frac{\epsilon }{c_2}.\)
Let \(r\in (0,\frac{\epsilon }{\lambda \mathcal {H}^1(\Sigma )})\)  be such that
\(|\varphi (x,\nu ) - \varphi (y,\nu )|<\frac{\epsilon }{\mathcal {H}^1(\Sigma )}\) 
whenever \(|x-y|<4r.\)  Since \(\Sigma \)  is Lipschitz, by Vitaly Covering Lemma we can find an at most countable family \(\lbrace Q_{r_j,\nu _{\Sigma }(x_j)}(x_j)\rbrace _{j\ge 1}\)  of disjoint open squares such that \(x_j\in \Sigma ,\)  \(r_j\in (0,r),\)  \(\Sigma \cap Q_{r,\nu _{\Sigma }(x_j)}(x_j)\)  is a graph in \(\nu _\Sigma (x_j)\) -direction,
\(\Sigma \)  crosses two opposite sides of each \(Q_{r,\nu _{\Sigma }(x_j)}(x_j)\)  parallel to \(\nu _\Sigma (x_j)\)  and
\(\mathcal {H}^1\Big (\Sigma ^{\prime }\setminus \bigcup _j \overline{Q_{r_j,\nu _{\Sigma }(x_j)}(x_j)}\Big )=0.\) 
Note that \(\sum \limits _j r_j< \mathcal {H}^1(\Sigma ).\) 
For each \(j\)  define
\(\Sigma _j:=(\Sigma \cap \overline{Q_{r,\nu _{\Sigma }(x_j)}(x_j)}) + \rho _j\nu _\Sigma (x_j),\) 
where \(\rho _j\in (0,r_j)\)  is such that \(\Sigma _j\)  still connects two vertical sides of \(Q_{r,\nu _{\Sigma }(x_j)}(x_j)\)  and \(\sum _j \rho _j <\frac{\epsilon }{2c_2}.\)  Let \(U_j\)  be the
open set whose boundaries are \(\Sigma _j,\)  \(\Sigma \cap \overline{Q_{r,\nu _{\Sigma }(x_j)}(x_j)}\)  and two vertical sides of \(Q_{r,\nu _{\Sigma }(x_j)}(x_j).\)  Note that \(\lbrace U_j\rbrace _j\)  is a countable family of pairwise disjoint open sets.
Let \(B_2:=B_1\setminus \overline{\cup _j U_j}\)  and \(v_2:=v_1\Big |_{B_2\cup S}.\)  Then using the localized version of \(\mathcal {S}\)  we get
\(\mathcal {S}(B_2,v_2) \le & \mathcal {S}(B_1,v_1-u_0; \Omega \setminus \overline{\cup _j U_j}) + \sum \limits _{j} \Big (\int _{\Sigma _j} \varphi (x,\nu _\Sigma (x))d\mathcal {H}^1 + 2c_2 \rho _j\Big ).\) 

By (REF ) and the definition of \(\Sigma _j\)
\(\int _{\Sigma _j} \varphi (x,\nu _\Sigma (x))d\mathcal {H}^1 \le \int _{\Sigma \cap Q_{r,\nu _{\Sigma }(x_j)}(x_j)} \varphi (y,\nu _\Sigma (y))d\mathcal {H}^1 + \frac{\epsilon \mathcal {H}^1(\Sigma \cap Q_{r,\nu _{\Sigma }(x_j)}(x_j))}{\mathcal {H}^1(\Sigma )}.\) 
Thus summing this inequality in \(j\)  and using pairwise disjointness of \(Q_{r,\nu _{\Sigma }(x_j)}(x_j)\)  and (REF ) we get
\(\sum \limits _j \int _{\Sigma _j} \varphi (x,\nu _\Sigma (x))d\mathcal {H}^1 \le \int _{\Sigma ^{\prime }} \varphi (y,\nu _\Sigma (y))d\mathcal {H}^1 + \frac{\epsilon \mathcal {H}^1(\Sigma ^{\prime })}{\mathcal {H}^1(\Sigma )}.\) 
Using the definition of \(\Sigma ^{\prime }\)  we obtain
\(\sum \limits _j \int _{\Sigma _j} \varphi (x,\nu _\Sigma (x))d\mathcal {H}^1 \le \int _{J_{v_1}} \varphi (y,\nu _\Sigma (y))d\mathcal {H}^1+ 2\epsilon .\) 
Inserting this in (REF ) and using the inequality \(\sum _j \rho _j <\frac{\epsilon }{2c_2}\)  we get
\(\mathcal {S}(B_2,v_2) \le \mathcal {S}(B_1,v_1-u_0; \Omega \setminus \overline{\cup _j U_j}) + \int _{J_{v_1}} \varphi (y,\nu _\Sigma (y))d\mathcal {H}^1+ 3\epsilon \le \mathcal {S}(B_1,v_1) + 3\epsilon .\) 
Then by the nonnegativity of the elastic energy, for \((B_2,v_2)\)  we get
\(\widetilde{\mathcal {F}}(B_2,v_2) \le \widetilde{\mathcal {F}}(B_1,v_1) + 3\epsilon .\) 
Notice that by our construction \(\Sigma \cap J_{v_2}\)  is \(\mathcal {H}^1\) -negligible, hence by Lemma REF  \(v_2\in H^1_\mathrm {loc}({\rm Int}{(B_2\cup S\cup \Sigma )}).\)
Finally we estimate the volume contribution of \(B_2.\)  Since \(U_j\subset Q_{r,\nu _{\Sigma }(x_j)}(x_j)\)  and
\(r_j\le r<\frac{\epsilon }{\lambda \mathcal {H}^1(\Sigma )},\)  using \(\sum _j r_j <\mathcal {H}^1(\Sigma )\)  we get
\(|B_1\setminus B_2| \le \sum \limits _j |U_j| \le \sum \limits _j r_j^2 \le r\sum \limits _j r_j < \frac{\epsilon }{\lambda }.\) 
Therefore,
\(\widetilde{\mathcal {F}}^{\lambda }(B_1,v_1) \ge \widetilde{\mathcal {F}}^{\lambda }(B_2,v_2) - 4\epsilon .\) 
Step 2: Let \(\lbrace E_i\rbrace _{i\ge 1}\)  be all open connected components of \(B_2\)  (recall that \(B_2\)  is open). We remove all sufficiently small connected components of \(B_1.\)  Using the localized versions of \(\mathcal {S}\)  and \(\mathcal {W}\)  we have
\(\mathcal {W}(B_2,v_2-u_0;\Omega ) = \sum \limits _{i\ge 1} \mathcal {W}(E_i,v_2-u_0;\Omega ).\) 
Since \(\partial E_i\cap \partial E_j\subset B_2^{(1)}\cap \partial B_2\)  and \(\varphi (x,\cdot )\)  is even,
\(\mathcal {S}(B_2,v_2;\Omega ) = \sum \limits _{i\ge 1} \mathcal {S}(E_i,v_2;\Omega ).\) 
Hence, there exists \(N_1\in \mathbb {N}\)  such that the set
\(B_3:=\cup _{i=1}^{N_1} E_i\) 
satisfies
\({\begin{array}{c}\mathcal {S}(B_2,v_2;\Omega ) +\mathcal {W}(B_2,v_2-u_0;\Omega )+ \epsilon >\mathcal {S}(B_3,v_2;\Omega )+\mathcal {W}(B_3,v_2-u_0;\Omega ),\\0 \le |B_2| - |B_3| < \frac{\epsilon }{\lambda }.\end{array}}\) 
Thus,
\(\mathcal {F}^{\lambda }(B_2,v_2) > \mathcal {F}^{\lambda }(B_3,v_3) - 2\epsilon ,\) 
where \(v_3:=v_2\Big |_{B_3}.\)
Step 3: Let \(\lbrace F_j\rbrace _{j\ge 1}\)  be all connected components of \(\Omega \setminus \overline{B_3}\)  such that \(\partial F_j\subset \partial B_3\)  (hence, \(F_i\)  are holes in \(B_3\) ). We fill in all sufficiently small holes. Since \(\mathcal {S}(B_3,v)<+\infty ,\)  there exists \(N_2\ge 1\)  such that
\(\sum \limits _{i>N_2} \mathcal {S}(F_i,v_3;\Omega ) + \sum \limits _{i>N_2} \mathcal {W}(F_i,v_3-u_0;\Omega ) <\epsilon ,\quad \sum \limits _{i>N_2} |F_i|<\frac{\epsilon }{\lambda }.\) 
Then the set \(B_4:= B_3\cup (\cup _{i>N_2} F_i)\)  and the function \(v_4:=v_3\chi _{B_2\cup S} + u_0\chi _{\cup _{i>N_2} F_i}\)  satisfies
\(\mathcal {F}^{\lambda }(B_3,v_3)>\mathcal {F}^{\lambda }(B_4,v_4) - 2\epsilon .\) 
By construction, \(\overline{\partial ^* B_4}\)  has at most \(N_1 + N_2\)  connected components.
Step 5: Finally we construct \((A_\epsilon ,u_\epsilon )\in \mathcal {C}_{n_\epsilon }\)  satisfying (REF ) for some \(n_\epsilon \in \mathbb {N}.\)  Let \(B_5:=\mathrm {Int}(\overline{B_4}).\)  Since \(B_4\)  can have finitely many “substantial” holes \(B_5\cap \partial B_4=\emptyset .\)  In particular, if we extend \(v_4\)  arbitrarily to the set \(B_4^{(1)}\cap \partial B_4\)  and denote the extension by \(v_5,\)  then \(v_5\in GSBD^2({\rm Int}{(B_5\cup S\cup \Sigma )})\)  and \(J_{v_5}=S_{v_4}^{B_4}\)  up to a \(\mathcal {H}^1\) -negligible set, where \(S_u^A\)  is defined in (REF ).
Since \(v_5=v_4\)  a.e. in \(B_5,\)  by (REF )-(REF )
\(\int _{B_5\cup S} x)e(v_5):e(v_5) = \mathcal {W}(B_4,v_4) \le \widetilde{\mathcal {F}}(B_4,v_4) + c_2\mathcal {H}^1(\Sigma ) \le C+ 9\epsilon ,\) 
where \(C:=\max \lbrace 1,\inf _{\widetilde{\mathcal {C}}}\widetilde{\mathcal {F}}\rbrace \)  is independent of \(\epsilon .\)
By [43]} there exists \(u_\epsilon \in SBV^2({\rm Int}{(B_5\cup S\cup \Sigma )})\cap L^\infty ({\rm Int}{(B_5\cup S\cup \Sigma )})\)  such that \(J_{u_\epsilon }\)  is contained in a union \(\Gamma \)  of finitely many closed connected pieces of \(C^1\) -curves in \({\rm Int}{(B_5\cup S\cup \Sigma )},\)  \(u_\epsilon \in W^{1,\infty }({\rm Int}{(B_5\cup S\cup \Sigma )})\)  and
\(\int _{B_5\cup S} |e(u_\epsilon ) - e(v_5)|^2 dx \le \frac{\epsilon }{4(C+11\epsilon )(\Vert _\infty +1)}\) 
and
\(\mathcal {H}^1(J_{u_\epsilon }\Delta J_{v_5})<\frac{\epsilon }{2c_2}.\) 
Since \(J_{v_5}\subset B_5,\)  we can assume that the squares \(\lbrace Q_j\rbrace _{j\ge 1}\)  of Vitali cover in [43]} satisfies \(Q_j\subset \subset B_5.\)  Therefore, we may assume that \(\Gamma \subset \overline{B_5}.\)  Since \(\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits \Gamma \)  is regular, we may extract finitely many intervals of \(\Gamma \)  whose union \(\Gamma ^{\prime }\)  still covers \(J_{u_\epsilon }\)  and satisfies \(\mathcal {H}^1(\Gamma ^{\prime }\setminus J_{u_\epsilon })<\frac{\epsilon }{2c_2}.\) 
Now we define
\(A_\epsilon :=B_5\setminus \overline{\Gamma ^{\prime }}.\) 
Recall that both \(\Sigma \cap J_{v_5}\)  and \(\Sigma \cap J_{e-\epsilon }\)  are \(\mathcal {H}^1\) -negligible.
By the definition of \(B_5\)  and \(\Gamma ^{\prime },\)  there exists \(n_\epsilon \in \mathbb {N}\)  such that \((A_\epsilon ,u_\epsilon )\in \mathcal {C}_{n_\epsilon }.\)  By the definition of \(\widetilde{\mathcal {S}},\)  \(B_5\)  and \(v_5\)  as well as by (REF ) we have
\(\widetilde{\mathcal {S}}(B_4,v_4) = & \int _{\Omega \cap \partial ^* B_5} \varphi (x,\nu _{B_5})d\mathcal {H}^1 + 2\int _{B_5\cap J_{v_5}} \varphi (x,\nu _{J_{v_5}})d\mathcal {H}^1 + \int _{\Sigma \cap \partial ^*B_5} \beta d\mathcal {H}^1\nonumber \\\ge & \int _{\Omega \cap \partial ^* B_5} \varphi (x,\nu _{B_5})d\mathcal {H}^1 + 2\int _{B_5\cap J_{u_\epsilon }} \varphi (x,\nu _{J_{u_\epsilon }})d\mathcal {H}^1 + \int _{\Sigma \cap \partial ^*B_5} \beta d\mathcal {H}^1 - \epsilon .\) 

Thus, by the definition of \(A_\epsilon \)  and \(\Gamma ^{\prime }\)
\( \widetilde{\mathcal {S}}(B_4,v_4) \ge & \int _{\Omega \cap \partial ^* A_\epsilon } \varphi (x,\nu _{A_\epsilon })d\mathcal {H}^1 + 2\int _{A_\epsilon ^{(1)}\cap \Gamma ^{\prime }} \varphi (x,\nu _{\Gamma ^{\prime }})d\mathcal {H}^1 + \int _{\Sigma \cap \partial ^*A_\epsilon } \beta d\mathcal {H}^1 - 2\epsilon \nonumber \\= & \mathcal {S}(A_\epsilon ,u_\epsilon ) - 2\epsilon .\) 

Moreover, using the relations \(|A_\epsilon \Delta B_4|=0\)  and \(v_4=v_5\)  a.e. in \(B_5\)  and Cauchy-Schwartz inequality for nonnegative symmetric forms we obtain
\( \mathcal {W}(A_\epsilon ,u_\epsilon ) \le & \mathcal {W}(B_4,v_4) + 2\int _{B_5\cup S} x)e(u_\epsilon ):(e(u_\epsilon ) - e(v_5))\nonumber \\\le & \mathcal {W}(B_4,v_4) + 2\Big (\int _{B_5\cup S} x)e(u_\epsilon ):e(u_\epsilon )dx\Big )^{1/2}\times \nonumber \\&\times \Big (\int _{B_5\cup S} x) (e(u_\epsilon ) - e(v_5)):(e(u_\epsilon ) - e(v_5))dx\Big )^{1/2}.\) 

Similarly,
\(& \int _{B_4\cup S} x)e(u_\epsilon ):e(u_\epsilon )dx\\\le & \mathcal {W}(B_4,v_4) + 2\Big (\mathcal {W}(B_4,v_4)\Big )^{1/2}\Big (\int _{B_5\cup S} x) (e(u_\epsilon ) - e(v_5)):(e(u_\epsilon ) - e(v_5))dx\Big )^{1/2}\\\le & (C+9\epsilon ) + 2\sqrt{(C+9\epsilon )\Vert _\infty }\,\,\Vert e(u_\epsilon ) - e(v_5)\Vert _{L^2} \le C+10\epsilon ,\) 

where in the last inequality we used (REF ).
Therefore, by (REF ) and again by (REF )
\(\mathcal {W}(A_\epsilon ,u_\epsilon ) \le & \mathcal {W}(B_4,v_4) + 2\sqrt{(C+10\epsilon )\Vert _\infty }\,\,\Vert e(u_\epsilon ) - e(v_5)\Vert _{L^2} \le \mathcal {W}(B_4,v_4) +\epsilon .\) 

Now combining (REF ) and (REF ) as well as using \(|B_5|=|A_\epsilon |\)  we get
\(\widetilde{\mathcal {F}}^{\lambda }(B_4,v_4) \ge \mathcal {F}^{\lambda }(A_\epsilon ,u_\epsilon ) - 3\epsilon .\) 
Since \((A_\epsilon ,u_\epsilon )\in \mathcal {C}_{n_\epsilon },\)  by (REF ), (REF ), (REF ), (REF ) and (REF ) we get
\(\inf \limits _{(A,u)\in \widetilde{\mathcal {C}}} \,\widetilde{\mathcal {F}}(A,u) + 12 \epsilon \ge \mathcal {F}(A_\epsilon ,u_\epsilon ),\) 
and (REF ) follows.
Proposition REF  implies that the configuration \((A,u)\)  given by Proposition REF  is a volume-constraint minimizer of \(\widetilde{\mathcal {F}}\)  in \(\widetilde{\mathcal {C}}.\)
Proposition 5.2 
Let \((A,u)\in \widetilde{\mathcal {C}}\)  be given by Proposition REF . Then
\((\mathrm {Int}(A),u)\)  is a minimizer of \(\widetilde{\mathcal {F}}\)  in \(\widetilde{\mathcal {C}}\)  under the volume constraint \(|A|=\mathtt {v}.\)  Moreover, let \(\lambda _0\)  be as in Proposition REF  and let \((B,v)\in \widetilde{\mathcal {C}}\)  be any volume-constraint minimizer of \(\widetilde{\mathcal {F}}.\)  Then \((B,v)\)  is a minimizer of \(\widetilde{\mathcal {F}}^\lambda \)  for all \(\lambda \ge \lambda _0,\)  where
\(\widetilde{\mathcal {F}}^{\lambda }(B,v):=\widetilde{\mathcal {F}}(B,v) + \lambda \big ||B| - \mathtt {v}\big |,\quad (B,v) \in \widetilde{\mathcal {C}},\quad \lambda >0.\) 


Note that since \(|\mathrm {Int}(A)\Delta A|=0\)  and \((\mathrm {Int}(A),u)\in \widetilde{\mathcal {C}},\)  by Propositions REF , REF  and  REF
\(\widetilde{\mathcal {F}}(\mathrm {Int}(A),u) = \inf \limits _{(B,v)\in \mathcal {C}}\, \mathcal {F}(B,v) = \inf \limits _{(B,v)\in \widetilde{\mathcal {C}}} \widetilde{\mathcal {F}}(B,v) = \inf \limits _{(B,v)\in \mathcal {C}} \widetilde{\mathcal {F}}^{\lambda }(B,v)\) 
for all \(\lambda \ge \lambda _0.\) 
Thus, \((\mathrm {Int}(A),u)\)  is a minimizer of both \(\widetilde{\mathcal {F}}\)  and \(\widetilde{\mathcal {F}}^{\lambda _0}.\)  The same is true for every minimizer \((B,v)\)  of \(\widetilde{\mathcal {F}}.\)
Theorem 5.3 (Density estimates for minimizers of \(\widetilde{\mathcal {F}}^\lambda \) ) 
Given \(\lambda >0,\)  let \((A,u)\in \widetilde{\mathcal {C}}\)  be any minimizer of
\(\widetilde{\mathcal {F}}^{\lambda }(\cdot ,\cdot )\)  in \(\widetilde{\mathcal {C}}\)  and let \(\xi \in \mathbb {R}^2\)  be such that for the function
\(\widetilde{u}:=u\chi _{A\cup S} + \xi \chi _{\Omega \setminus A}\) 
one has \(\Omega \cap \partial ^*A \subset J_{\widetilde{u}}.\)  Then for any \(x\in \Omega \)  and \(r\in (0,\mathrm {dist}(x,\partial \Omega )),\)
\(\frac{\mathcal {H}^1( Q_r(x)\cap J_{\widetilde{u}})}{r} \le \frac{16c_2 + 4\lambda }{c_1}.\) 
Moreover, there exist \(\varsigma ^*=\varsigma ^*\in (0,1)\)  and \(R^*>0\)  not depending on \((A,u)\)  with the following property. If \(x\in \Omega \)  belongs to the closure \(J^c_{\widetilde{u}}\)  of the set \(\lbrace y\in \Omega \cap J_{\widetilde{u}}:\,\theta _*(J_{\widetilde{u}},y)>0\rbrace ,\) 
then
\(\frac{\mathcal {H}^1( Q_r(x)\cap J_{\widetilde{u}} )}{r}\ge \varsigma ^*\) 
for any square \( Q_r(x)\subset \subset \Omega \)  with
\(r\in (0,\min \lbrace R^*,\mathrm {dist}(x,\partial \Omega )\rbrace ),\)  and if \(x\in \Omega \)  belongs to the closure \(S_u^c\)  of \(\lbrace x\in S_u^A:\,\theta _*(S_u^A,x)>0\rbrace ,\)  then
\(\frac{\mathcal {H}^1( Q_r(x)\cap S_u^A)}{r}\ge \varsigma ^*\) 
for any \(r\in (0,\min \lbrace R^*,\mathrm {dist}(x,\overline{\partial ^*A})\rbrace .\) 
In particular,
\(\mathcal {H}^1(\Omega \cap (J_{\widetilde{u}}^c\setminus J_{\widetilde{u}})=\mathcal {H}^1(\mathrm {Int}(A^{(1)})\cap (S_u^c\setminus S_u^A)=0.\) 


As in Remark REF  \((A,u)\)  is a minimizer of \(\widetilde{\mathcal {F}}^{\lambda }\)  if and only if
\((A,u+u_0)\)  minimizes the \( \widehat{\widetilde{\mathcal {F}}^{\lambda }}(\cdot ):=\widetilde{\mathcal {F}}^{\lambda }(\cdot -u_0).\)  Thus, we can introduce the following localized version of \(\widetilde{\mathcal {F}}\)  in open subsets \(O\)  of \(\Omega \)  which does not see the substrate:
\(\widetilde{\mathcal {F}}(B,v;O):=\widetilde{\mathcal {S}}(B;O) + \mathcal {W}(B,v; O)\) 
where
\(\widetilde{\mathcal {S}}(B,v;O):= \int _{O\cap \partial ^*B} \varphi (y,\nu _B)d\mathcal {H}^1+ 2\int _{O\cap B^{(1)}\cap \partial B\cap S_v} \varphi (y,\nu _B)d\mathcal {H}^1,\) 
the \(\mathcal {W}(\cdot ;O)\)  is given as in (REF )
and \(S_v^A\)  is defined as in (REF ).
Then the minimality of \((A,u)\)  implies that \((A,u+u_0)\)  is a quasi-minimizer of \(\widetilde{\mathcal {F}}(\cdot ;O)\) 
in \(O,\)  namely,
\(\widetilde{\mathcal {F}}(A,u +u_0;O) \le \widetilde{\mathcal {F}}(B,v;O) + \lambda _0|A\Delta B|\) 
whenever \((B,v)\in \widetilde{\mathcal {C}}\)  with \(A\Delta B\subset \subset O\)  and \(\mathrm {supp}\,(u+u_0-v)\subset \subset O.\) 
Now the proof of the existence of \(\varsigma ^*\)  and \(R^*\)  satisfying (REF ) and (REF ) runs along the same lines of the proof of Theorem REF  for \(m=\infty ,\)  therefore, we do not repeat it here. Note that \(\varsigma ^*\)  and \(R^*\)  depend only on \(c_i\)  and \(\lambda .\)
Let \(A_{\circ }:=\mathrm {Int}(A^{(1)}).\)  We claim that
\(\partial A_{\circ } = \overline{\partial ^*A}.\) 
Indeed, note that \(A^{(1)}\setminus A_{\circ }\subset \partial A^{(1)} =\overline{\partial ^*A},\)  where in the equality we used \(\overline{\partial ^*A} = \overline{\partial ^*A^{(1)}}=\partial A^{(1)}\)  see e.g., [45]}. Thus, \(A_{\circ }\)  is also equivalent to \(A,\)  and hence, \(\partial ^*A_{\circ }=\partial ^*A=\partial ^*A^{(1)}.\)  In particular, \(\partial A^{(1)}=\overline{\partial ^*A_{\circ }}\subset \partial A_{\circ }.\)  On the other hand, assume that there exists \(x\in \partial A_{\circ }\setminus \partial A^{(1)}.\)  Since \(\partial A^{(1)}\)  is closed, there exists \(r>0\)  such that \(\overline{B_r(x)}\cap \partial A^{(1)}=\emptyset .\)  Hence, either \(B_r(x)\subset \mathrm {Int}(A^{(1)})=A_{\circ }\)  or \(\overline{B_r(x)}\cap \overline{A^{(1)}}=\emptyset .\)  Since \(A_{\circ }\)  is open and \(x\in \partial A_{\circ },\)  the inclusion \(B_r(x)\subset A_{\circ }\)  is not possible. On the other hand, since \(\overline{A_{\circ }}\subset \overline{A^{(1)}}\)  and \(x\in \partial A_{\circ },\)  the relation \(\overline{B_r(x)}\cap \overline{A^{(1)}}=\emptyset \)  is also not possible. Thus, \(\partial A_{\circ }\subseteq \partial A^{(1)}.\)
To prove (REF ) we fix \(\Omega ^{\prime }\subset \subset \Omega .\) 
We claim that \(\widetilde{u}\big |_{A_{\circ }}\)  is a minimizer of Griffith functional \(\mathcal {G}:GSBD^2({\rm Int}{(A_{\circ }\cup S\cup \Sigma )}) \rightarrow \mathbb {R},\)
\(\mathcal {G}(v):=\int _{A_{\circ }\cap J_{v}} \varphi (x,\nu _{J_v}) d\mathcal {H}^1 + \int _{A_{\circ }} x)e(v):e(v)dx\) 
with Dirichlet boundary condition \(v=\widetilde{u}=u\)  in \(A_{\circ }\setminus \Omega ^{\prime }.\)  Indeed, for every \(v\in GSBD^2(A_{\circ })\)  with \(\widetilde{u}=v\)  in \(A_{\circ }\setminus \Omega ^{\prime }\)  we define \(B:=A_{\circ }\setminus \overline{J_v}.\)  Then \((B,v)\in \widetilde{\mathcal {C}}\)  and by the minimality of \((A,u)\)
\(\mathcal {G}(u) - \mathcal {G}(v) =\widetilde{\mathcal {F}}(A,u) - \widetilde{\mathcal {F}}(B,v)\le 0.\) 
Since \(S_v^B = J_{\widetilde{u}\big |_{A_{\circ }}}\)  up to a \(\mathcal {H}^1\) -negligible set, (REF ) directly follows from the density estimates for the jump set of Griffith minimizers (see e.g. [46]}) with possibly smaller \(\varsigma ^*\in (0,1)\)  and \(R^*>0.\)
Finally, we prove (REF ) only for \(S_u^A,\)  the other being similar. Let \(\Gamma :=\lbrace x\in S_u^A:\, \theta _*(S_u^A,x)>0\rbrace .\)  Note that \(S_u^c=\bar{\Gamma }.\)
We claim that
\(\mathcal {H}^1(A_{\circ }\cap (\bar{\Gamma }\setminus \Gamma ))=0.\) 
Indeed, let \(\mu :=\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits \Gamma .\)  Then \(\mu (\bar{\Gamma }\setminus \Gamma )=0.\)  By the regularity of \(\mu ,\)  for every \(\epsilon >0\)  there exists an open set \(U\subset \mathbb {R}^2\)  such that \(L:=A_{\circ }\cap (\bar{\Gamma }\setminus \Gamma ) \subset U\)  and \(\mu (U)=\mathcal {H}^1(U \cap \Gamma )<\epsilon .\)  Note that \(\bar{\Gamma }\subset \overline{\lbrace y\in \Omega \cap J_{\widetilde{u}}:\,\theta _*(J_{\widetilde{u}},x)>0\rbrace },\)  where \(\widetilde{u}\)  is given by Theorem REF . Hence, for (REF ) holds for all points of \(\bar{\Gamma }.\)  By the definition of the closure, and Vitali Covering Lemma we can find at most countable pairwise disjoint family \(\lbrace \overline{B_{r_i}(x_i)}\rbrace _i\)  of closed balls \(\overline{B_{r_i}(x_i)}\)  with \(x_i\in A_{\circ }\cap \Gamma ,\)  \(r_i\le \min \lbrace R^*,\epsilon ,\mathrm {dist}(x,\partial \bar{A})\rbrace \)  such that \(A_{\circ }\cap (\bar{\Gamma }\setminus \Gamma ) \subset \cup _i \overline{B_{5r_i}(x_i)}.\)  Without loss of generality we may assume that \(B_{r_i}(x_i)\subset U.\)  Since \(Q_{r_i/\sqrt{2}}(x_i)\subset B_{r_i}(x_i)\subset Q_{r_i}(x_i),\)  from the definition of Hausdorff premeasure, (REF ) and disjointness of \(\lbrace B_{r_i}(x_i)\rbrace \)  as well as the choice of \(U\)  we obtain
\(\begin{aligned}\mathcal {H}_{10\epsilon }(A_{\circ }\cap (\bar{\Gamma }\setminus \Gamma ))\le & \sum \limits _{i\ge 1} 2\pi (5r_i) \le \frac{10\pi \sqrt{2}}{\varsigma ^*} \sum \limits _{i\le 1} \mathcal {H}^1(Q_{r_i/\sqrt{2}}(x_i) \cap \Gamma ) \\=& \frac{10\pi \sqrt{2}}{\varsigma ^*}\, \mathcal {H}^1\big (\cup _i Q_{r_i/\sqrt{2}}(x_i) \cap \Gamma \big ) \le \frac{10\pi \sqrt{2}}{\varsigma ^*}\, \mathcal {H}^1\big (U \cap \Gamma \big ) <\frac{10\pi \sqrt{2}\epsilon }{\varsigma ^*}.\end{aligned}\) 
Now letting \(\epsilon \rightarrow 0\)  we get (REF ).
In the following proposition we construct a “regular” minimizer of \(\mathcal {F}\)  starting from a minimizer of \(\widetilde{\mathcal {F}}\)  in \(\widetilde{\mathcal {C}}.\)
Proposition 5.4 
Given \(\lambda >0,\)  let \((A,u)\in \widetilde{\mathcal {C}}\)  be any minimizer of \(\widetilde{\mathcal {F}}^\lambda .\)  Then there exists \((A^{\prime },u^{\prime })\in \mathcal {C}\)  such that
\(\widetilde{\mathcal {F}}(A,u) = \mathcal {F}(A^{\prime },u^{\prime }).\) 
Moreover:
(1)
\(A^{\prime }\)  is open, \(\theta _*(S_{u^{\prime }}^{A^{\prime }},x)>0\)  for all \(x\in S_{u^{\prime }}^{A^{\prime }},\)  \(|A\Delta A^{\prime }|=0\)  and \(u\chi _{A\cup S} = u^{\prime }\chi _{A^{\prime }\cup S}\)  a.e. in \(\Omega \cup S;\)
(2)
the closure of \(A^{\prime }\cap \partial A^{\prime }\)  coincide with \(\overline{S_{u^{\prime }}^{A^{\prime }}}\)  and \(\mathcal {H}^1(\overline{S_{u^{\prime }}^{A^{\prime }}}\setminus S_{u^{\prime }}^{A^{\prime }})=0;\)
(3)
let \(\varsigma *\)  and \(R^*\)  be given by Theorem REF . Then
\(\frac{\mathcal {H}^1(Q_r(x)\cap \partial A^{\prime })}{r} \le \frac{16c_2 + 4\lambda _0}{c_1}\) 
for every square \(Q_r(x)\subset \Omega \) 
and
\(\frac{\mathcal {H}^1(Q_r(x)\cap \partial A)}{r} \ge \varsigma *\) 
for every \(Q_r(x)\subset \Omega \)  with for any \(x\in \partial A^{\prime }\)  and \(r\in (0,R^*).\)


Let \(\Gamma :=\lbrace x\in S_u^A:\,\theta _*(S_u^A,x)>0\rbrace \)  and let
\(A^{\prime }:=\mathrm {Int}(A^{(1)})\setminus \bar{\Gamma }\) 
and let \(u^{\prime }\)  be the extension of \(u\)  to \(A^{\prime }\cap \partial A\)  such that \(J_{ u^{\prime }} =S_u^A.\)  Note that \(A^{\prime }\)  is open, \(|A^{\prime }\Delta A|=0\)  and \(u\chi _{A\cup S} = u^{\prime }\chi _{A^{\prime }\cup S}\)  a.e. in \(\Omega \cup S.\) 
Moreover, \(S_{u^{\prime }}^{A^{\prime }}\subset \Gamma ,\)  and by (REF )
all points of \(\Omega \cap \bar{\Gamma }\)  satisfy uniform lower density estimates, hence, \(\theta _*(S_{u^{\prime }}^{A^{\prime }},x)>0\)  for any \(x\in S_{u^{\prime }}^{A^{\prime }}.\)
We claim that \(A^{\prime }\in \mathcal {A}.\)  Indeed, let \(\widetilde{u}\)  be given as in Theorem REF . By definition
\(\Omega \cap J_{\widetilde{u}}^c = \Omega \cap \partial A^{\prime }\quad \text{and}\quad \partial A^{\prime } \subset J_{\widetilde{u}}^c\cup \Sigma ,\) 
where \(J_{\widetilde{u}}^c\)  is the closure of the set \(\lbrace x\in J_{\widetilde{u}}:\,\theta _*(J_{\widetilde{u}},x)>0\rbrace .\)  Since \(J_{\widetilde{u}}\)  is \(\mathcal {H}^1\) -rectifiable, so is \(J_{\widetilde{u}}^c\)  in view of (REF ).
Therefore, \(\partial A^{\prime }\)  is \(\mathcal {H}^1\) -rectifiable, i.e., \(A^{\prime }\in \mathcal {A}.\) 
Note that by construction \(\mathcal {H}^1(A^{\prime }\cap J_{\widetilde{u}})=0\)  hence, by Lemma REF  \(\widetilde{u}\in H_\mathrm {loc}^1(A^{\prime })\)  and, since \(u=\widetilde{u}\)  a.e. in \(A^{\prime }\)  it follows that \(u^{\prime }\in H_\mathrm {loc}^1(A^{\prime }).\)
Since \(|A\Delta A^{\prime }|=0\)  and \(u^{\prime }=u\)  a.e. in \(A^{\prime },\)  it follows that
\(\mathcal {W}(A,u) =\mathcal {W}(A^{\prime },u^{\prime }).\) 
Moreover, by the definition of \(\Gamma \)  and \(S_u^A,\)
\(|\mathcal {S}(A^{\prime },u^{\prime }) - \widetilde{\mathcal {S}}(A,u)| = \int _{\mathrm {Int}(A^{(1)})\cap (\bar{\Gamma }\setminus S_u^A)} \varphi (x,\nu _\Gamma )d\mathcal {H}^1 \le c_2\mathcal {H}^1(\mathrm {Int}(A^{(1)})\cap (\bar{\Gamma }\setminus \Gamma )) =0,\) 
where in the last equality we used (REF ). Finally, (REF ) and (REF ) follows from (REF ) and density estimates of Theorem REF .
Hence, \((A^{\prime },u^{\prime })\)  satisfies all properties stated in the proposition.
Now we are ready to prove the existence of global minimizers of \(\mathcal {F}.\)
First we prove the assertion for \(\mathcal {G}=\mathcal {F}.\)
Let \((A_m,u_m)\in \mathcal {C}_m\)  be a minimizer of \(\mathcal {F}\)  satisfying the volume constraint \(|A_m|=\mathtt {v}\)  and
let \((A_{m_h},u_{m_h}),\)  and \(A\)  and \(u\) 
be as in Proposition REF .
By (REF ), (REF ) and (REF ) we have
\(\inf \limits _{(B,v)\in \mathcal {C},\,|B|=\mathtt {v}} \mathcal {F}(B,v) = \lim \limits _{h\rightarrow +\infty } \mathcal {F}(A_{m_h},u_{m_h}) \ge \widetilde{\mathcal {F}}(\mathrm {Int}(A),u).\) 
Since \( |\mathrm {Int}(A)|=\mathtt {v},\)  by Propositions REF  and  REF
\(\inf \limits _{(B,v)\in \mathcal {C},\,|B|=\mathtt {v}} \mathcal {F}(B,v)=\inf \limits _{(B,v)\in \widetilde{\mathcal {C}},\,|B|=\mathtt {v}} \widetilde{\mathcal {F}}^{\lambda _0}(B,v)=\widetilde{\mathcal {F}}^{\lambda _0}(\mathrm {Int}(A),u)=\widetilde{\mathcal {F}}(\mathrm {Int}(A),u),\) 
hence, \((\mathrm {Int}(A),u)\)  is a minimizer of \(\widetilde{\mathcal {F}}^{\lambda _0}\)  in \(\widetilde{\mathcal {C}}.\)  Then by Proposition REF  there exists \((A^{\prime },u^{\prime })\in \mathcal {C}\)  such that
\(\widetilde{\mathcal {F}}(\mathrm {Int}(A),u) = \mathcal {F}(A^{\prime },u^{\prime }),\) 
and hence, in view of (REF ), \((A^{\prime },u^{\prime })\)  is a solution to (REF ).
The proof of the second assertion (i.e., the existence of \(\lambda _1\)  for which the set of minimizers in \(\mathcal {C}\)  of both \(\mathcal {F}\)  and \(\mathcal {F}^\lambda \)  coincide for all \(\lambda \ge \lambda _1\) ) can be done using the first one and also following the arguments of [47]} and [5]}. Without loss of generality we assume that \(\lambda _1\ge \lambda _0,\)  where \(\lambda _0\)  is given by Proposition REF .
Now we prove Theorem REF  for \(\mathcal {G}=\widetilde{\mathcal {F}}.\)  We have already shown above that the configuration \((\mathrm {Int}(A),u)\)  given by Proposition REF  solves the minimum problem (REF ) with \(\mathcal {G}=\widetilde{\mathcal {F}}.\)  In view of (REF ) every volume-constraint minimizer of \(\widetilde{\mathcal {F}}\)  also minimizer of \(\widetilde{\mathcal {F}}^\lambda \)  for all \(\lambda \ge \lambda _1.\)  To prove the converse assertion, we fix any minimizer \((A,u)\in \widetilde{\mathcal {C}}\)  of \(\widetilde{\mathcal {F}}^\lambda \)  for \(\lambda \ge \lambda _1.\)  By Proposition REF  there exists \((A^{\prime },u^{\prime })\in \mathcal {C}\)  such that \(|A^{\prime }|=|A|\)  and \(\mathcal {F}(A^{\prime },u^{\prime })=\widetilde{\mathcal {F}}(A,u).\)  By the first part of the proof and (REF ) we know that
\(\inf \limits _{(B,v)\in \mathcal {C}}\mathcal {F}^\lambda (B,v) = \inf \limits _{(B,v)\in \mathcal {C},|B|=\mathtt {v}}\mathcal {F}(B,v) = \inf \limits _{(B,v)\in \widetilde{\mathcal {C}}}\widetilde{\mathcal {F}}^\lambda (B,v) =\mathcal {F}^\lambda (A^{\prime },u^{\prime }).\) 
Hence, \((A^{\prime },u^{\prime })\)  is the minimizer of \(\mathcal {F}^\lambda .\)  Since \(\lambda \ge \lambda _1\)  according to the first part of the proof, \( |A^{\prime }|=\mathtt {v}.\)  Hence, \(|A|=\mathtt {v}\)  and \((A,u)\)  minimizer of (REF ).
We are ready now to study the properties of the minimizers of \(\mathcal {F}\)  in \(\mathcal {C}\)  provided by Theorem REF .
First we properties (1)-(4) the assertion for \(\mathcal {G}=\widetilde{\mathcal {F}}.\)
Consider any solution \((A,u)\in \widetilde{\mathcal {C}}\)  of (REF ).
By Proposition REF  there exists a \((A^{\prime },u^{\prime })\in \mathcal {C}\)  with \(A^{\prime }\)  defined as in (REF ), such that the properties (1)-(4) hold except the conditions \(\mathcal {H}^1(\partial A\Delta \partial A^{\prime })=0\)  and \(\mathcal {H}^1(S_u^A\Delta S_{u^{\prime }}^{A^{\prime }})=0\)  of (1). To prove these two equations it is enough to observe that
\(0=|\mathcal {F}(A,u) - \mathcal {F}(A^{\prime },u^{\prime })| =2\int _{A^{(1)}\cap (\partial A\Delta \partial A^{\prime })} \varphi (x,\nu _A)d\mathcal {H}^1\) 
and
\(0=|\widetilde{\mathcal {F}}(A,u) - \widetilde{\mathcal {F}}(A^{\prime },u^{\prime })| =2\int _{A^{(1)}\cap (S_u^A\Delta S_{u^{\prime }}^{A^{\prime }})} \varphi (x,\nu _A)d\mathcal {H}^1.\) 
Now we assume that \(\mathcal {G}=\mathcal {F}\)  and let \((A,u)\in \mathcal {C}\)  be a solution to (REF ). Since \((A,u)\in \widetilde{\mathcal {C}},\)  by Proposition  REF
\(\inf \limits _{(B,v)\in \widetilde{\mathcal {C}},|B|=\mathtt {v}} \widetilde{\mathcal {F}}(B,v) = \mathcal {F}(A,u) \ge \widetilde{\mathcal {F}}(A,u).\) 
Therefore, \((A,u)\)  is also a volume-constraint minimizer of \(\widetilde{\mathcal {F}}.\)  Thus, applying first part of the proof we establish that \((A^{\prime },u^{\prime })\in \mathcal {C}\)  satisfies (1)-(4).
Finally, notice that if \(E\subset A^{\prime }\)  is a connected component of \(A^{\prime }\)  with \(\mathcal {H}^1(\partial E\cap \Sigma \setminus J_u)=0,\)  then for \((A^{\prime },v)\)  with \(v=u^{\prime }\chi _{(A\cup S)\setminus E} + (u_0 +a)\chi _E,\)  where \(a\)  is any rigid displacement,
we have
\(\mathcal {S}(A^{\prime },u^{\prime }) \ge \mathcal {S}(A^{\prime },v)\) 
and
\(\mathcal {W}(A^{\prime },u^{\prime }) \ge \mathcal {W}(A^{\prime },v),\) 
where in (REF ) equality holds if and only of \(u^{\prime }=u_0+a\)  in \(E.\)  Therefore, by the minimality of \((A^{\prime },u^{\prime })\)  it follows that \(u^{\prime }=u_0+a\)  in \(E.\)  It remains to prove
\(|E|\ge 4\pi \,\Big (\frac{c_1}{\lambda _0}\Big )^2.\) 
Consider the competitor \((A^{\prime }\setminus E,u^{\prime })\in \mathcal {C}.\)  By minimality and Theorem REF , \(\mathcal {F}^{\lambda _1}(A^{\prime },u^{\prime }) \le \mathcal {F}^{\lambda _1}(A^{\prime }\setminus E,u^{\prime }),\)  so that by (REF ) and the additivity of the surface energy,
\(\mathcal {S}(E,u^{\prime }) \le \lambda _1 |E|.\) 
Then by (REF ) and the isoperimetric inequality in \(\mathbb {R}^2\)
\(\lambda _1 |E| \ge c_1 \mathcal {H}^1(\partial E) \ge c_1\sqrt{4\pi } |E|^{1/2}.\) 
Hence, (REF ) follows.

appendix


Lemma 1.1 
Let \(K\subset \mathbb {R}^2\)  be such that \(\mathcal {H}^1(K)<+\infty \)  and let \(\lbrace E_t\rbrace _{t\in }\)  be a family of sets parametrized by \(t\in \)  such that
\(\mathcal {H}^1(K\cap E_t\cap E_s)=0\) 
and \(\mathcal {H}^1(K\cap E_t)>0.\)  Then \(\)  is at most countable.

The proof runs along the lines of the proof of [45]}.
For \(j\in \mathbb {N}\)  let \(_j\subset \)  be the set of all \(t\in \)  such that \(\mathcal {H}^1(K\cap E_t)>\frac{1}{j}.\)  Then by (REF ) \(_j\)  cannot contain more than \(j\mathcal {H}^1(K)\)  elements. Since \(=\cup _j _j,\)  the set \(\)  is at most countable.
Lemma 1.2 
Let \(U\subset \mathbb {R}^n\)  be a connected bounded open set and \(u\in GSBD^2(U)\)  be such that \(\mathcal {H}^{n-1}(J_u)=0.\)  Then \(u\in H_\mathrm {loc}^1(U).\)

Indeed, for \(r>0\)  let \(Q:=x_0+(-r,r)^n\subset U\)  be any cube centered at \(x\in U\)  and let \(0<\theta ^{\prime \prime }<\theta ^{\prime }<1.\)  For shortness, write \(Q^{\prime }:=x_0+(-\theta ^{\prime } r,\theta ^{\prime }r)^n\)  and \(Q^{\prime \prime }:=x_0+(-\theta ^{\prime \prime }r,\theta ^{\prime \prime }r)^n.\)  By [50]} (see also [1]}) there exists a \(\mathcal {L}^2\) -measurable set \(\omega \subset Q^{\prime }\)  and a rigid displacement \(a:\mathbb {R}^2\rightarrow \mathbb {R}^2\)  such that \(|\omega |\le c_* r\mathcal {H}^{n-1}(J_u)=0\)  and
\(\int _{Q^{\prime }} |u -a|^{\frac{2n}{n-1}}dx = \int _{Q^{\prime }\setminus \omega } |u -a|^{\frac{2n}{n-1}}dx\le c_*r^2 \Big (\int _Q |e(u)|^2\Big )^{\frac{n}{n-1}},\) 
where \(c_*\)  depends only on \(n.\)  Hence, \(u\in L_\mathrm {loc}^{\frac{2n}{n-1}}(Q).\)  Next, fix any mollifier \(\rho _1\in C^\infty (B_r(0))\)  with \(\rho _\epsilon \in C_c^\infty (B_{(\theta ^{\prime \prime } - \theta ^{\prime })\epsilon }),\)  where \(\rho _\epsilon (x):=\rho _1(x/\epsilon ),\)  \(\epsilon \in (0,r).\) 
By [50]} there exists \(\bar{p}>0\)  depending on \(n\)  and \(\epsilon \)  such that
\(\int _{Q^{\prime \prime }} |e(u*\rho _\epsilon ) - e(u)*\rho _\epsilon |^2 dx \le c \Big (\frac{\mathcal {H}^{n-1}(J_u)}{r^{n-1}}\Big )^{\bar{p}} \int _Q |e(u)|^2dx =0,\) 
where \(c\)  depends on \(n,\)  \(\rho _1\)  and \(\epsilon .\) 
Hence,
\(e(u*\rho _\epsilon ) = e(u)*\rho _\epsilon \quad \text{a.e. in $Q^{\prime \prime }.$}\) 
Recall that \(u*\rho _\epsilon \in C^\infty (Q^{\prime \prime }).\)  Since \(e(u)\in L^2(Q),\)  \(e(u)*\rho _\epsilon \in C^\infty (Q^{\prime \prime })\cap L^2(Q^{\prime \prime })\)  in particular, \(e(u*\rho _\epsilon ) \in C^\infty (Q^{\prime \prime })\cap L^2(Q^{\prime \prime }).\)  By Poincaré-Korn inequality \(u*\rho _\epsilon \in H^1(Q^{\prime \prime }).\)  Since \(e(u)*\rho _\epsilon \rightarrow e(u)\)  in \(L^2(Q^{\prime \prime })\)  as \(\epsilon \rightarrow 0,\)  in view of (REF ) there exists \(\epsilon _0>0\)  such that
\(\Vert e(u*\rho _\epsilon )\Vert _{L^2(Q^{\prime \prime })} \le \Vert e(u)\Vert _{L^2(Q^{\prime \prime })} +1\quad \text{for all $\epsilon \in (0,\epsilon _0).$}\) 
Moreover, by Poincaré-Korn inequality for any \(\epsilon \in (0,\epsilon _0)\)  there exists a rigid displacement \(a_\epsilon \)  such that
\(\Vert  u*\rho _\epsilon - a_\epsilon \Vert _{H^1(Q^{\prime \prime })} \le C \Vert e(u*\rho _\epsilon )\Vert _{L^2(Q^{\prime \prime })} \le C(\Vert e(u)\Vert _{L^2(Q^{\prime \prime })} + 1),\) 
where \(C\)  is the Poincaré-Korn constant for a cube.
Thus, the family \(\lbrace u*\rho _\epsilon \rbrace _\epsilon \)  is uniformly bounded in \(H^1(Q^{\prime \prime })\) . Since \(u*\rho _\epsilon \rightarrow u\)  in \(L^2(Q^{\prime \prime }),\)  there exists a rigid displacement \(a\)  such that \(a_\epsilon \rightarrow a\)  in \(L^2(Q^{\prime \prime }).\)  Then \(u*\rho _\epsilon - a_\epsilon \)  weakly converges to \(u-a\)  in \(H^1(Q^{\prime \prime }),\)  i.e., \(u-a\in H^1(Q^{\prime \prime }).\)  Since \(a\)  is linear and \(\theta ^{\prime \prime }\)  is arbitrary, \(u\in H_\mathrm {loc}^1(Q).\)  Now covering \(U\)  with finitely many cubes of edgelength \(2r\)  we get \(u\in H_\mathrm {loc}^1(U).\)

Acknowledgments
Sh. Kholmatov acknowledges support from the Austrian Science Fund (FWF)
project M 2571-N32. P. Piovano acknowledges support from the Austrian Science Fund (FWF) project P 29681, from the Vienna Science and Technology Fund (WWTF) together with the City of Vienna and Berndorf Privatstiftung through Project MA16-005, and from BMBWF through the OeAD-WTZ project HR 08/2020.
